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Portal:Fire and Emergency Management

Welcome to the School of Fire and Emergency Management
Today's fire and emergency services have evolved considerably. While steeped in a rich history, the new fire service tradition is one of diversity of skills and knowledge. The vision of the School of Fire and Emergency Management is to be a portal for training and education; as well, to provide a resource for related research and data. Our approach is titled Open Source Learning. To learn more about it, read the paper About Open Source Learning Project by Michael D. Finney. Building upon this foundation, education and learning will be the key to the future of the fire and emergency services. Upon this foundation, we begin our journey. The beauty of this community is the ability to collaborate at all levels and to share that knowledge base with everyone. In short, the success is based on the ability to share what we know and to build upon it.

Current Fire and Emergency Management Students If you are a current student, looking for information or a program then have a browse. If we don't have what you are looking for, then we hope you will find it, add it here and wait for others to review what you've added. We're naturally all volunteers, but there's more chance of finding what you're searching for if you talk about it. If you would like to see something added, please click here.

Brand New If you are brand new to the School of Fire and Emergency Management, then take a moment to familiarize yourself with your surroundings. Much like new apparatus or equipment, familiarization builds confidence and skills. As well, the community will constantly be growing and developing, so taking time now to get one's navigational bearings will save a lot of frustration down the road.

Academia If you are an academic, instructor, or training officer, then we welcome you to edit, add and evolve the site. Your knowledge and expertise will add greatly to the integrity and morale of the School, as well as adding confidence to end users. It's also quite fun. You can add to current courses or create your own.

To All The most important thing, is that if you have some information you can share - then share it! We welcome all contributions.

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Training Materials
Airport Rescue Fire Fighting

Driver/Operator

Emergency Medical Services

Fire and Life Safety Educator

Fire Fighter

Fire Inspector

Fire Instructor

Fire Investigator

Fire Officer

Hazardous Materials

Technical Rescue

Public Safety Telecommunicator

Terrorism

Wildland Firefighter

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Departmental Message Board

 * Fire and Emergency Management Department Message Board
 * Who is Who in the Fire and Emergency Management Department

 * Strategic Planning for the School

 * 9-11 Memorial

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Library

 * The Library
 * Fire Service Wikipedia Portal
 * Chemistry Wikipedia Portal
 * Medicine Wikipedia Portal
 * Wiktionary
 * | Open Fire Academy related to fire service online learning.

 * Current Research Projects

This is the gateway for information about the courses and training in general. We hope that its size and scope will increase dramatically as participation in the School increases. We welcome input from all.

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Current Research and Project News
''Post here if you have a research project or proposal you are working on. Also, if there is a course you would like to develop and would like to know if others are in need of it or are working on it.''

 * Leadership I (under Fire Officer) is complete and ready for use...please provide feedback and suggestions on the discussion page of the main course page.

 * We are in need of volunteers to begin translating some of the projects into other languages. If you are interested, please email Mfinney for more information of where to begin.

title: Portal:Fire and Emergency Management


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User:Shekira08/sandbox

The Islamic marriage contract is a major part of the marriage process that involves prenuptial agreement.It must have at least two people that are witnessing the ceremony going on.If there are not two people that witness their the two people can be witnesses to their own ceremony.

source: wikimedia

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: User:Shekira08/sandbox


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User:Jimerb/sandbox

Main Heading
This is a test. Richard Mingus

Signature Formatting
 Jimerb

Usage share of web browsers

title: User:Jimerb/sandbox


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Pan American Table Tennis Cup

The Pan American Table Tennis Cup is an annual table tennis competition held in the Americas since 2017. It consists of men's and women's singles events. Only 16 invited players and no more than 2 players per association are allowed to participate in each event. Since its inception, it is recognised as the qualification event for the Table Tennis World Cup. The competitions are organized as a joint effort of the Latin American Table Tennis Union and the Northern American Table Tennis Union. The event is sanctioned by the International Table Tennis Federation (ITTF).

title: Pan American Table Tennis Cup


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Herbert von Reyl

Herbert von Reyl (28 April 1898 – 11 June 1937) was an Austrian painter. His work was part of the painting event in the art competition at the 1936 Summer Olympics.

title: Herbert von Reyl


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Robyn van Nus

Robyn van Nus (born 10 December 1982) is an Australian sport shooter. She competed in two events at the 2008 Summer Olympics and the 2012 Summer Olympics; the Women's Air Rifle, 10 metres event where she finished 44th overall in 2008 and 45th in 2012, and the Women's Small-Bore Rifle, Three Positions, 50 metres event where she finished 40th overall in 2008 and 41st in 2012.

Personal
van Nus was born on 10 December 1982 in Attadale, Western Australia. She attended St Josephs Primary School and St. Norbert College in Western Australia. In 2010, she enrolled at Central Queensland University where she is working on Bachelor of Exercise and Sports Science.

, she lives in Sydney.

, van Nus was 164 cm tall and weighed 60 kg, unchanged from 2008. She has chronic fatigue syndrome. Her mother also represented Australia in shooting.

Shooting
van Nus is a sport shooter, specialising in the rifle. She took up the sport around 1998 as a way to deal with chronic fatigue syndrome with her mother serving as her first coach. Petr Kůrka became her coach in 2009. She was a member of the Melbourne International Shooting Club. In 2012, she was a member of the Kalamunda Smallbore Rifle Club. She has a shooting scholarship with the Western Australian Institute of Sport.

van Nus competed in the 2008 Summer Olympics in the Women's Air Rifle, 10 metres event where she finished 44 overall and in the Women's Small-Bore Rifle, Three Positions, 50 metres event where she finished 40th overall. Going into the 2008 Games, her inclusion was challenged by Susannah Smith because van Nus was selected over Smith by having a one-point lead. She competed at the 2010 Commonwealth Shooting Championship, earning a bronze in the 10m air rifle event with a score of 494.2 (394). In the women's 10-metre air rifle pairs, she and partner Aletha Sedgman finished sixth. She had some difficulties with the competition because "her gun and the rest of her equipment was lost in transit to Delhi."

In 2011, van Nus was the number one ranked Australian woman in her discipline. In Brisbane in a July 2011 at national tournament, she broke her own women's air rifle Australian record when she shot 397 out of 400. In October 2011, she participated in the Australia Cup Final in Sydney, but withdrew mid-competition because of a back injury after only shooting nine of her forty shots. She was selected to represent Australia at the 2012 Summer Olympics in shooting. London were her second Games.

title: Robyn van Nus


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Baghshan

Baghshan is a village in Zeberkhan Rural District of the Central District of Zeberkhan County, Razavi Khorasan province, Iran.

At the 2006 National Census, its population was 2,769 in 797 households, when it was in the former Zeberkhan District of Nishapur County. The following census in 2011 counted 2,958 people in 940 households. The latest census in 2016 showed a population of 3,127 people in 992 households. It was the most populous village in its rural district.

After the census, Zeberkhan District was separated from Nishapur County in the establishment of Zeberkhan County, which was divided into two districts of two rural districts each, with Qadamgah as its capital and only city.

title: Baghshan


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Talk:Rocky Mountain High

Album Cover
If anyone here is good at uploading, the Rocky Mountain High album cover is available to upload (I think) at John Denver.com. Check it out. Editor19841 18:07, 22 April 2006 (UTC)

Shouldn't the album cover be that of the original album Rocky Mountain High? It just makes more sense to use the cover that matches the audio sample. The cover from Laserlight was released in 1997. The audio was recording in the early '90's. —Preceding unsigned comment added by <IP_ADDRESS> (talk) 14:56, 23 August 2010 (UTC)

Banned in the USA ?
I once heard about some weirdo action group who tried to have this song banned because it had the word "high" in it and therefore, in their opinion, promoted the use of cannabis.
 * It is in the article now. Full lyrics of the song apparently include "friends around the campfire and everybody's high", much more explicit than the title (which, to be honest, I thought was a high school at first because I was 12 years old, and had only heard the lyrics over a loudspeaker and couldnt make out most of the words).  —  Soap  —  13:10, 3 April 2011 (UTC)

External links modified
Hello fellow Wikipedians,

I have just modified 1 one external link on Rocky Mountain High. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
 * Corrected formatting/usage for http://www.aspentimes.com/article/20071013/NEWS/71012028/0/FRONTPAGE

Cheers.—cyberbot II  Talk to my owner :Online 15:06, 1 April 2016 (UTC)

title: Talk:Rocky Mountain High


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User:Olivier 99

My name is Olivier; As a new memeber of this wonderfull Wikipedia Project, i will have to learn a lot about this encyclopedia; after that i hope to contribute to it.

I live in Lennik, a small town near Brussels (Belgium) I study history at the University of Ghent (first bachelor)

title: User:Olivier 99


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Lanjiang Road station

Lanjiang Road Station, is a station of Line 4 of Wuhan Metro. It entered revenue service on December 28, 2014. It is located in Hanyang District.

title: Lanjiang Road station


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User:Gdo01/Boards

On Notice
 * Kaname Tōsen - they had to screw up the only black guy in Bleach (manga)
 * UTC - Because of you I need to have extra clocks on Wikipedia
 * Mainstream rappers - Alternative hip hop is true to its origins unlike mainstream rap which caters to middle-class white guys while providing bad influences to young African-Americans

Dead to Me
 * Beyoncé Knowles - what ever happened to being a survivor and an independent women?
 * Stacy Ferguson - My Humps, if you've heard the song you know how dumb and meaingless it is
 * Mariah Carey - her songs are about needing a man, badly needing a man
 * Gwen Stefani - Left No Doubt to cater to horny hip-hop-wannabe men and because of her slave women, the Harajuku Girls
 * Nelly Furtado - after several years of good alternative music she comes out with Promiscuous (song)
 * Geraldo Rivera - puts the name Geraldo to shame

Never Existed to Me
 * Flavor Flav - Flavor of Love is a modern day minstrel show

title: User:Gdo01/Boards


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Gregorio Martínez Sacristán

Gregorio Martínez Sacristán' (19 December 1946 – 20 September 2019) was a Spanish Roman Catholic bishop.

Martínez Sacristán was born in Villarejo de Salvanés, Spain and was ordained to the priesthood in 1971. He served as bishop of the Roman Catholic Diocese of Zamora in Spain from 2006 until his death in 2019.

title: Gregorio Martínez Sacristán


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User:Afrosade/sandbox

Request review at WP:AFC
AFRO SADE MUSIC
 * Indented line

"a new sensation of roots african music

'''AFROSADE'[[File:

Example.jpg
File:Example.jpg|Caption1 Afrobeat and Juju music are deeply rooted in NIGERIA, so is Sade Emmanuel AKA “AFROSADE”. Born in Lagos of Yoruba mother and Ibo father, raised in England and USA. AfroSade is one of the few Nigerian female singers, if not the only one who fused root Fela Kuti style Afrobeat, with Juju Genre. She was exposed to Fela Kuti music by her father who was a great fan. she was discovered in New York city by Kaleta of ( Zozo Afrobeat), an Ex-Fela Kuti’s cohort and now her music producer..

title: User:Afrosade/sandbox


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Talk:Mount Vernon, Illinois

"Local Sports" section removed
I removed the entire "Local Sports" section because it doesn't belong here - it belongs in an article about the local high school instead. Furthermore, it was poorly sourced and full of speculation or personal opinion. The part about a local flag football league was not notable.

Crime section
is misleading and/or untrue.

"According to NeighborhoodScout, "With a crime rate of 77 per one thousand residents, Mount Vernon has one of the highest crime rates in America compared to all communities of all sizes - from the smallest towns to the very largest cities. One's chance of becoming a victim of either violent or property crime here is one in 13. Within Illinois, more than 99% of the communities have a lower crime rate than Mount Vernon."[23]"

The source does not list where it gets it data from - if you compare Mt. Vernon to other cities, big and small. The statistics provided don't make sense. Joliet and Carbondale, for example are listed on the "2015 Top 25 Most Dangerous Cities in Illinois" based on FBI crime data, but is ranked significantly safer than Mt. Vernon. (http://herald-review.com/news/state-and-regional/most-dangerous-cities-in-illinois/collection_a8e253ac-baf2-56c7-9dbb-506ca232f58b.html) Unless a valid source can be offered. I think this section should be removed.

Statistics at http://www.usa.com/rank/illinois-state--crime-index--city-rank.htm?yr=6000&dis=50&wist=&plow=&phigh= disprove the claim that Mount Vernon has one of the highest crime rates in America.

— Preceding unsigned comment added by <IP_ADDRESS> (talk) 15:49, 10 June 2015 (UTC)

Added facts directly from FBI UCR rather than from any site that does not provide detailed source information

I had originally added the NeighborhoodScout information, but it is as vague as usa.com in that both use their own algorithms to rank or rate areas. The FBI UCR warns about this when accessing their information stating that it is easily misunderstood and/or skewed toward one view. To avoid that bias I am creating a table which holds the numbers from the UCR only. It is only comprised of their rate presented in the tables as crimes per 100,000 population. This information is extrapolated by the FBI from reported information by the policing body, in this case MVPD. I do not intend to delete the usa.com information as that is perhaps best left to someone more experienced than I am at this time.

Dirtvoyles (talk) 19:21, 13 July 2015 (UTC)

City motto
Corrected city motto as found on the history page of Jefferson county IL website. "The King City" would more likely go under Nickname, although citizens often do refer to the city as Mousetown to mock many in leadership. Maybe research is needed to determine the correct usage.Dirtvoyles (talk) 15:42, 4 August 2015 (UTC)

Founding date
I find information about the founding of the city by Zadok Casey around 1817, but information on the city website seems to assure that it was established as a city in 1819 and then the county seat shortly after that action. I would appreciate any help in finding an undisputed source.Dirtvoyles (talk) 15:42, 4 August 2015 (UTC)

External links modified
Hello fellow Wikipedians,

I have just added archive links to 1 one external link on Mount Vernon, Illinois. Please take a moment to review my edit. If necessary, add after the link to keep me from modifying it. Alternatively, you can add to keep me off the page altogether. I made the following changes:
 * Added archive https://web.archive.org/20150217104348/http://www.nationalrailway.com/nre-mt-vernon to http://www.nationalrailway.com/nre-mt-vernon

Cheers. —cyberbot II  Talk to my owner :Online 14:03, 29 August 2015 (UTC)

Can anyone help with citations for 19th - 21st century?
I've done a decent job I think organizing the page as best I can and tracked down a few resources for citations but much of the information seems irrelevant and more of a family tree. Perhaps I'm biased.

It would be nice to have input from editors with decent experience to advise and guide progress. — Preceding unsigned comment added by Dirtvoyles (talk • contribs) 17:35, 26 June 2018 (UTC)

title: Talk:Mount Vernon, Illinois


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Latin/Lesson 2

Spelling and Pronunciation
The Latin alphabet, on which the English alphabet is based, has mostly the same letters as the English alphabet, except that it has no <k> or <w>, and that in its original form, it lacked <j>, which only some modern texts use, and <u>. Many European languages use the Latin alphabet as the basis for their own alphabet.

Latin pronunciation has varied somewhat over the course of its long history, and there are some differences between Old Latin, spoken in the Roman Republic, Classical Latin, spoken in the Roman Empire, Medieval Latin, spoken in the Middle Ages, and Ecclesiastical Latin, spoken in the Catholic Church. This text focuses on the pronunciation of Classical Latin.

Note that Latin, as written by the Romans, did not have <j>, <k>, <u>, <w>, or macrons over vowels (the lines indicating that vowels are long), although they did sometimes mark long vowels with apices (e.g. <ó> for /oː/); macrons are used today as pronunciation guides and do not necessarily need to be written. /w/, /ʊ/, and /uː/ were all represented with <v>. Modern texts often use <v> for /w/, <u> for /ʊ/, and <ū> for /uː/. In some modern texts (this Wikibook not included), <j> is used for /j/.

Declension Tables
The following tables will be both referenced and explained in all of the following sections, and hence are placed here.

Note that nouns in the 3rd declension nominative can have any ending, hence why none is given in bold.

Grammar Part 1: Nouns and Their Role in Sentences
Nouns in Latin are inflected, which means that endings (also known as suffixes or suffices) are appended to the end of the stem to denote these things:
 * 1) Number (whether the noun is singular or plural)
 * Case of the noun (role of the noun in the sentence)
 * Gender (the gender of the word - one of masculine, feminine, or neuter)

Most nouns in English can be modified to indicate number (cat versus cats), and many pronouns can be modified to indicate case (who versus whose) or gender (he versus she, his versus hers). Case is especially important in Latin as meaning cannot be determined by word order as it can be in English, but purely by word endings, or "inflection". Indeed, the words in a Latin sentence can appear in almost any order with little change in meaning. Two sentences with the word orders "Sam ate the orange" and "The orange ate Sam" could potentially mean the same thing in Latin, though the spellings of "orange" and "Sam" would have to change slightly to denote which was the subject (the one eating) and which was the object (the one being eaten).

It is important to note here that although the genders of many words make sense (for example, "puella", meaning a girl, is feminine) many are simply assigned and hold no real meaning. Luckily, as you will find, the gender can often be determined by the spelling of the word (words ending in "us" are almost always masculine, and words ending in "a" are almost always feminine). For many words, however, you will simply have to memorize their gender.

Adjectives themselves must match the number, case, and gender of the noun (be it a substantive or a pronoun) they modify. If a noun is nominative singular feminine (see case table below), then the adjective describing it must also be nominative singular feminine. If the noun is accusative plural masculine, then the adjective must be accusative plural masculine. This will be expanded on in the Adjectives section below. The advantage of this system is that adjectives do not need to be adjacent to their respective nouns, as one would be able to tell which noun they modify by which noun they appear to agree with.

Declension
All substantives are part of one of 5 categories, called declensions. A substantive is a stem, modified by adding a declension suffix. Each declension has a set of standard suffixes that indicate case and number. Usually gender is indicated by the suffix, although there are many exceptions. Therefore, you must memorize the gender of every substantive you learn.

By familiarizing yourself with the above tables, you could deduce that originally the suffix indicating number, case, and gender was the same for every noun. However, as the language developed, nouns with a common stem formed declensions and sounds changed. Similar processes happen continually over time, even today.

The above tables allow you to familiarize yourself with the existence of each declension, though by no means are you expected to memorize it now. Nonetheless, you will have to memorize it as you are formally introduced to individual cases and declensions in future lessons. Because of its introductory purpose, it is considerably simplified and incomplete, and therefore should not be used as a reference in the future.

Adjectives are also classed into declensions which must match the declension of the noun they describe:
 * 1) 1st/2nd declension adjectives...
 * 2) ...Use 1st declension suffixes from the substantive declension table when describing feminine nouns.
 * 3) ...Use 2nd declension masculine suffixes from the above table when describing masculine nouns.
 * 4) ...Use 2nd declension neuter suffixes (not found in the above table) when describing neuter nouns.
 * 5) 3rd declension adjectives behave as 'i' stem substantives unless specified. An 'i' stem substantive is one where the stem word ends in the letter 'i.'   Masculine and Feminine suffixes (which are the same) will be used if describing masculine and feminine nouns, and Neuter suffixes will be used when describing neuter nouns.

Pronouns are not part of any declension, as they are all irregular, and simply have to be memorized.

Case
Cases (Latin: casus) determine the role of the noun in the sentence in relation to other parts of the sentence.

There are six cases, Nominative, Genitive, Dative, Accusative, Ablative, and Vocative. Vocative case (Lesson 3), can be considered a sort of miniature case, generally not being accepted as a true one. Additionally, some nouns have a vocative case, which will be covered later. As nominative and accusative are the most basic, these will be taught first (the rest will be covered in later lessons).

Gender
All substantives, including inanimate objects, have a particular gender (genera), which is either masculine, feminine, or neuter.

For example, Vir, "a man," is masculine. Marītus, "a husband," is also masculine. Puella, "a girl," is feminine. Māter, "a mother," is feminine. Even inanimate objects are assigned gender, including all the moons, stars, trees, tools, and so forth. Logic will give you little help in determining what the genders of inanimate objects are, and with many nouns memorization is required. Luckily, for many nouns, the spelling of the word indicates the gender.

Certain rules may be utilized to determine the gender of an inanimate substantive. Declension is a good indication of gender, especially for 1st and 2nd declension substantives. 1st declension substantives (substantives with an -a suffix) are usually feminine and second declension nouns (substantives with an -us suffix) are usually masculine or neuter. There are a few exceptions, and they will have to be learned. 3rd declension nouns can be either masculine, feminine or neuter (thus the gender will often have to be memorized). 4th declension nouns are usually masculine, sometimes neuter while 5th declension nouns are usually feminine.

Nouns undeclined, words which are not substantives but used as such, sentences used as substantives and the products of trees are generally neuter.

1st/2nd declension adjectives alternate the set of endings depending on the gender of the noun it describes (see the next section below). If the adjective describes a feminine noun, the adjective must use 1st declension endings, if the adjective describes a masculine noun, the adjective must use 2nd declension masculine endings, if the adjective describes a neuter noun, the adjective must use 2nd declension neuter endings.

3rd declension adjectives use the same set of endings for masculine and feminine nouns. However, a slightly different set of endings are used when describing neuter nouns.

Adjectives
As stated above, adjectives must match the gender, number, and case of the noun (be the noun a substantive, or a pronoun) they modify. Similar to the "Sam ate the orange" example above, if the adjective uses the wrong declension it could change the meaning of the sentence. For example, "The girl loves big trees," versus "the big girl loves trees" have different meanings. There are many occasions where logic cannot be used to determine the gender of inanimate objects, as genders are generally arbitrary when the noun has no literal gender. Furthermore, the declension of the noun, often determined by the spelling, can in turn be used to determine the gender, especially for the 1st and 2nd. However, this is never the case for the third declension, as the declension itself is not primarily assigned to any gender and the spelling of the nominative ("default") stem is random, leaving you with no hints.

A noun and its adjective must also be in the same case. Otherwise, it is impossible to tell which nouns pair up to their respective adjectives in a sentence, as the words in a Latin sentence can appear in any order. See the examples below.

Recapitulation

 * Declensions are used to categorize nouns in groups. There are 5 declensions total.
 * Each of the five declensions has a distinct set of endings which are appended to nouns of that declension.
 * The endings indicate the case and number when appended to the stem of a noun.
 * A substantive may use only the endings of the declension of which it is a part.
 * Each substantive has a predefined gender which almost never changes and is separate from the suffix.
 * Adjectives are a part of the 1st/2nd declension and 3rd declension.
 * Adjectives use the gender of the noun that they modify.

Therefore:
 * An adjective of the 1st/2nd declension uses 1st declension endings when describing a feminine noun, a 2nd declension masculine ending when describing a masculine noun, and 2nd declension neuter when describing a neuter noun.
 * An adjective of the 3rd declension uses the same set of endings when describing masculine and feminine nouns and another set of endings when describing neuter nouns. (Actually, there are 3-termination, 2-termination, and 1-termination 3rd declension adjectives. If the adjective is 3-termination, e.g., acer (f. sing.), acris (m. sing.), acer (n. sing.), acres (f. pl.), acres (m. pl.), or acria (n. pl.), then use the appropriate ending; if the adjective is 2-termination, then one termination will be masculine/feminine and the other neuter; if the adjective is 1-termination, the common form is used.)

Before you proceed to the next lesson, complete the exercises below so you will be able to apply this knowledge to Latin.

title: Latin/Lesson 2


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Calculus

Calculus uses methods originally based on the summation of infinitesimal differences.

It includes the examination of changes in an expression by smaller and smaller differences.

Resources

 * Calculus/Introduction
 * Definitions
 * Calculus I
 * Calculus II
 * Fundamental Calculus
 * /Quick Reference/

Osnabrück Calculus
Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I is a course for beginners at the University of Osnabrück. It covers logical foundation, sets, mappings, algebraic structures like fields and polynomials, the basics of analysis like sequences, continuity, differentiability, primitive functions and the basics of linear algebra like vector spaces, bases, linear maps, eigenvalues. While course has been taught many times, this is a textbook -- not an online course. The lectures dealing with calculus are as follows.
 * Contents • Logic and argumentation• Quantifiers and induction• Sets and mappings• Fields• Complex numbers• Polynomials• Approximation and convergence• Completeness• Series• Continuity• Intermediate value theorem• Exponential function• Trigonometry• Differentiability• Mean value theorem• The number • Taylor series• Integration• Fundamental theorem of calculus• Rules for integration

Lessons
Wikibooks has a well developed Calculus Book in 10 sections which this Wikiversity web course should mirror to maximize resources.

Other Resources

 * Calculus
 * Various Calculus Problems
 * Calculus of generating functions

Online Textbooks

 * Wikibooks Calculus
 * Prof Dale Hoffman. (2007). Contemporary Calculus: Part 1
 * R.Courant. Differential and Integral Calculus, Vol. I by Richard Courant
 * R.Courant. Differential and Integral Calculus, Vol. II by Richard Courant
 * Gilbert Strang. (1991). Calculus
 * Ray Mayer. (2007) Calculus I - Theoretical focus.

Hard-copy Textbooks

 * Spivak, Michael. (1994). Calculus ISBN<PHONE_NUMBER>
 * Spivak, Michael. (1965). Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus ISBN<PHONE_NUMBER>
 * Gilbert Strang. (1991). Calculus ISBN<PHONE_NUMBER>

Textbook Supplements

 * Elliott Mendelson. (1988). 3,000 Solved Problems in Calculus ISBN<PHONE_NUMBER>
 * Frank Ayes, Elliott Mendelson. (1999). Schaum's Outline of Calculus (Fourth Edition) ISBN<PHONE_NUMBER>
 * Solutions to selected exercises from Apostol's Calculus Vol. 1
 * Solutions to selected exercises from Calculus by Larson, Hostetler, Edwards

Wikipedia

 * Calculus
 * List of basic calculus topics
 * List of calculus topics
 * Stochastic calculus

title: Calculus


text:
User:Coraleetrigger/sandbox

North Texas Now is Denton's only daytime talk show.It airs weekdays at 9:00 am and 2:00 pm Central on North Texas Television (Charter 22, Verizon FiOS 46). The half-hour show premiered on February 15, 2011 and is currently hosted by Sazan Barzani Matthew Torres.

Production
North Texas Now (NTN) is Denton's only daytime talk show. It is shot multi-camera style, live-to-tape. Like all ntTV programming, North Texas Now is produced entirely by student volunteers.

Mission
The mission is use television and new media to to create thoughtful programming that informs, inspires, and entertains our campus and the North Texas community.

Regular Segments
Off The Highway - (Coming Season 4) Denton Highlight - (Season 1-present) Denton Highlights include: A Creative Art Studio, Museum on the Square, Mad World Records, La Di Da, Jupiter House, Denton County Independent Hamburger Company, Hickory Creek Lounge, Abby Underground, Vigne Wine, Simone Lounge. UNT Highlight - (Season 1-present) Highlighting UNT students, organizations, and departments. Segments include: UNT Theater's performance of Godspell, Student Money Management, Student Government Association, Pike Fraternity... Steve on the Streets - (Season 1-3) Segment host Steve Hendrix takes it to the streets in his segment. Shot with one photographer (handheld), Steve explores the Denton and DFW communities and asks lots of questions of people on the streets along the way. Segments include: "Speed Dating", "What's your pet peeve?", "Fry vs. Square" Music Now - (Season 1-present) Music Now highlights local and up and coming musicians. Segments guest include: Mark My Word, Acey's Ghost, and Nelly's Echo.

Season 4
North Texas Now Season 4 (Fall 2012) will premiere in September.

Season 3
North Texas Now Season 3 (Spring 2012) was hosted by Sazan Barzani and Matthew Torres and Executive Produced by Coralee Trigger. In its third season, the show celebrated its first live-audience show, its 50th episode, completed a spin-off pilot for a late-night talk show, its first weekend shoot, and 5 other on-location shoots (The Dallas Arboretum, Casa Manana, The Dallas Contemporary, The University of North Texas Library Mall, and The Shops at Legacy in Plano).

Season 2
North Texas Now Season 2 (Fall 2011) was hosted by Sazan Barzani and Andy Edwards and Executive Produced by Coralee Trigger. In its second season, the show completed 18 episodes. On-location episodes were shot at The Dallas Arboretum, Six Flags Over Texas (for Fright Fest), Discovery Gardens, and The Denton Square.

Season 1
North Texas Now Season 1 (Spring 2011) was hosted by Kristen Diaz and Adam Hasley and Executive Produced by Trey Friedrichs. In its inaugural season, the show became the first ntTV entertainment program to complete 23 episodes in one semester. Episodes were shot in The University of North Texas Student Union, with the exception of 1.19 and 1.20 which were shot on-location at The Dallas Arboretum. Frequent segments included "Cooking Now" and "Music Now."

Make A Difference
On April 15, 2011, the North Texas Now team recorded their first "Make A Difference" episode. It was an hour long special that highlighted kids, students, adults, and community leaders along with organizations around Denton and the Dallas-Fort Worth Metroplex who were making a difference in their communities. Guests included the creator/founder of TOMS shoes, Blake Mycoskie. Each season that followed has had its own "Make A Difference" episode.

Milestones
On April 15, 2011, NTN recorded its first "Make A Difference" episode. It aired on April 19.

On April 22, 2011, the show taped off-campus for the first time at The Denton Square. Episode 1.15: "Hip to be Square: Day" aired on April 26 and featured daytime activities around the Denton Square. Episode 1.16: "Hip to be Square: Night" aired on April 28 and featured Denton's nightlife.

On April 30, 2011, the show taped out of Denton for the first time at The Dallas Arboretum. Episode 1.19: "The Dallas Arboretum" aired on May 10, 2011 and episode 1.20: "A Summer Day in Dallas" aired on May 12, 2011.

On March 9, 2012, the NTN team left North Texas for the first time to shoot an episode in Austin during the first weekend of the 2012 SXSW Festival.

On April 6, 2012, the show had its first live studio audience. The episode was recorded in the University of North Texas student union courtyard.

On April 13, 2012, the NTN team shot a pilot for a late-night talk show, "NTN at Night with Brandon Becvar." It was renamed "Late Night at North Texas with Brandon Becvar" and is scheduled to premiere on North Texas Television in October 2012.

On April 20, 2012, NTN shot its 50th episode. The episode was recorded in the University of North Texas student union courtyard with a live studio audience.

Notable Guests
Tim Urban, American Idol Season 7 Finalist, Singer-Songwriter - episode 1.08 (interview and performance) Blake Mycoskie, Creator/Founder of TOMS Shoes - 1.13 (pre-recorded segment) The Dallas Zoo representatives and touring animals - episode 2.04 Matt Nordgren, star of Most Eligible Dallas - episode 2.08 (pre-recorded segment) Bonnie Blossman, star of Big Rich Texas- episode 2.11 (pre-recorded segment) The Twilight Saga: Breaking Dawn stars Nikki Reed, Ashley Greene, and Jackson Rathbone - episode 2.14 (pre-recorded segment on the red carpet at Gilly's Dallas) Imaj, Singer-Songwriter - episode 3.02 (interview and performance) Mike Krieger, co-founder of Instagram - episode 3.05 Kevin Hart, star of Think Like A Man - episode 3.07 (pre-recorded segment on the red carpet)

On-Location
The show began taping around The University of North Texas student union, but has since gone off-campus to shoot episodes around Denton and the DFW Metroplex.

On-Location Episodes
1.15: "Hip to be Square: Day" - The Denton Square 1.16: "Hip to be Square: Night" - The Denton Square 1.19: "The Dallas Arboretum" - The Dallas Arboretum 1.20: "A Summer Day in Dallas" - The Dallas Arboretum 2.06: "A Fall Day in Dallas" - The Dallas Arboretum 2.09: "Six Flags Fright Fest" - Six Flags Over Texas 2.13: "Discovery Gardens" - Discovery Gardens 2.17: "Best of Season 2" - The Denton Square (Holiday Tree Lighting Ceremony) 3.01: "Valentine's Day" - The Dallas Arboretum 3.02: "The Dallas Contemporary" - The Dallas Contemporary 3.03: "Casa Manana" - The Casa Manana Theater 3.04: "The Library Mall" - Outside on the Library Mall at The University of North Texas 3.05: "A SXSW Weekend in Austin" - Austin, TX 3.09: "Best of Season 3" - The Shops at Legacy in Plano

Notable On-Location Segments
1.20: "A Summer Day in Dallas" - Highland Park Pharmacy 1.20: "A Summer Day in Dallas" - JD's Chippery 2.03: "Wild Wild West" - Billy Bob's Texas 2.04: "Creepy, Crawly, Soft, and Fuzzy: The Animal Show!" - The Fort Worth Zoo 2.06: "A Fall Day in Dallas" - The Nasher Sculpture Center 2.06: "A Fall Day in Dallas" - Texas State Fair 2.06: "A Fall Day in Dallas" - Sixth Floor Museum 2.06: "A Fall Day in Dallas" - Texas Rangers Stadium 2.11: "Trends" - Rooster's Roadhouse 3.01: "Valentine's Day" - Dallas Fashion Week 3.05: "A SXSW Weekend in Austin" - The Austonian 3.05: "A SXSW Weekend in Austin" - Berry Austin 3.05: "A SXSW Weekend in Austin" - Ballet Austin 3.05: "A SXSW Weekend in Austin" - SXSW (unofficial coverage) 3.05: "A SXSW Weekend in Austin" - 6th Street 3.05: "A SXSW Weekend in Austin" - Austin, TX

Fun Facts
For episode 2.05: "Living The Green Life" the North Texas Now crew accepted the challenge of going vegetarian for the week leading up to the shoot.

title: User:Coraleetrigger/sandbox


text:
Introduction to Calculus/Limits

==Introduction to Limit andLimit processes are the basis of calculus. As opposed to algebra, where a variable is considered to have a fixed value (think of the solution of word problems, where there are one or more discrete answers), we allow a variable to change continuously and study how a function's value changes.==

Outline
This article addresses limits of functions of a single variable. It starts with an informal definition, discusses the basic properties of the limit operation, and progresses to the precise definition of limit. This rigorous definition is used to prove the earlier results, which were stated without proof. A number of examples of applying the definition are given, which helps develop facility with inequalities.

An Informal Definition
Consider the function $$y = 2x$$. As x increases, y increases. As we crank x up towards a number, say 100, y gets closer to the number 200.

Limits are concerned with what the value of a function (in this case, y) approaches as a variable it is based on (in this case, x) approaches a number (in this case, 100), not the actual value of the function when the variable equals the number. This is useful because not all functions are continuous and you may get a different result in the limit of y as you crank x down instead of up to that number.

Continuous Functions

 * Look at these two graphs. Notice that the top one is a single, unbroken curve, whereas the bottom one has many "jumps." Now the top graph is continuous and the bottom one is not. This is the basic idea of continuity for a function; a function that is not continuous will have jumps. Points themselves can also be continuous and not continuous. For example, the point where x equals -1/2 in the second graph is continuous because there are no jumps in that specific section. On the other hand, where x equals two, there is a jump, so the point is not continuous. Not a very formal definition, this jump thing! Because the jump thing is so informal, they invented a new way to phrase it using limits

Definition of a Continuous Function
Say we want to prove that the top function is continuous for all values between -3 and 3. (This is a closed interval, or a section of a function that includes the two endpoints. Likewise, an open interval is a section of a function that does not include the two endpoints. The closed interval between -3 and 3 includes -3 and 3; the open interval does not.) The first definition is that of continuity in an interval

we want to prove that the top function is continuous for all values between -3 and 3. (This is a closed interval, or a section of a function that includes the two endpoints. Likewise, an open interval is a section of a function that does not include the two endpoints. The closed interval between -3 and 3 includes -3 and 3; the open interval does not.) The first definition is that of continuity in an interval

 * Definition of continuity in a closed interval
 * A continuous closed interval is continuous at every point, including the left and right endpoints.

Not too complicated, eh? Now on to the definition of continuity at a point that is in a open interval; that is, not including the endpoints

 * Definition of continuity at a point in an open interval in a function
 * Let c be the x-value of the coordinate where we want to prove that the function is continuous. The function f is said to be continuous at the point c if the three following conditions hold:

 * $$f(c)$$ exists,
 * $$\lim_{x\to c} f(x)$$ exists, and
 * $$\lim_{x\to c} f(x)=f(c)$$

Now, when can the limit as x approaches c not exist? Just look at the second graph! If you try to find the limit at two from numbers larger than two, you get a result that does not equal the results from using numbers smaller than two! There is a way to write this mathematically, so we define new types of limits.

 * The mathematical way to write "The limit of f(x) as x approaches c created by using numbers larger than c" is $$\lim_{x\to c^+}f(x)$$
 * The mathematical way to write "The limit of f(x) as x approaches c created by using numbers smaller than c" is $$\lim_{x\to c^-}f(x)$$

Basic Theorems for Limit Operations
The limit operator satisfies linearity. That is, "the limit of sum, is the sum of the limits".

There are also basic rules for doing arithmetic with limits. They can be found in the calculus textbook for reference. Follow this link and study the limits. Convince yourself that these rules are intuitive. Also note that if f(a) is defined, and if f is a continuous function, then $$\lim_{x\to a} f(x)=f(a)$$.

Proofs
What does $$\lim_{x\to 5} (x^2)$$ equal?

Find the value of this expression for values close to 5:

Notice how, as we get closer to 5 from both sides, the value of the function,  x2  approaches 25. This may seem obvious, since 5 squared actually equals 25. In fact, the limit of  f(x)  as x approaches c in a continuous open interval is equal to  f(c)  if it is defined. However, this problem is merely to help you get acquainted to limits, before we go into limits about a very special quotient,  0/0  and other indeterminate forms, or expressions that cannot be determined by substituting c for x in a limit where x goes to c.

NOTE: Wikipedia is a very good reference on these limits; however, avoid going into the section on evaluating indeterminate limits, for this uses a statement called l'Hôpital's rule. This is a useful rule but is confusingly unnecessary until you know what a derivative is.

Notice that  0/0  by itself is meaningless; also notice that $$\lim_{x\to 0}(x/x)$$ equals one, since for all values near 0,  x/x=1 .

The main purpose of limits
You can find otherwise undefined expressions with limits. They allow you to use algebraic rules, even at values when the rules are false! For example, look at $$\lim_{x\to 0} (x^2/x)$$. What does this equal?

Let's use an algebraic rule that is true at all values of x besides zero. The rule states that $$x^2/x$$ equals  x  for all numbers beside 0. When we apply this rule to our old limit, we see that the limit is equivalent to $$\lim_{x\to 0} x$$, which is easily seen to be equal to 0.

For more on finding limits, see the calculus wikibook.

A More Precise Definition of Limit

 * For almost all purposes, the informal definition of a limit works very well; however, because of its vague wording, it is very dificult to use it in any sort of proof about limits. For proofs, the formal definition of a limit, from Wikibooks, is used instead.

Limits involving Polynomials
Finding the limit of a polynomial is a simple process. The easiest method of taking the limit of a polynomial is substitution. The value of a polynomial as x approaches a is equal to f(a). Consider the following example:

$$\lim_{x\rightarrow 3} x^3 + 2x^2 + 7 = (3)^3 + 2(3)^2 + 7 = 52 $$

Limits involving Trigonometric Functions
This is a unit circle. It has a radius of one unit, and its angles are measured in radians. Using this circle, we can prove that
 * $$\lim_{\theta\to 0} \,\ \frac{\sin(\theta)}{\theta}=1$$
 * Notice that this makes sense, since as theta approaches 0, arc DA becomes very close to being congruent to arc DC. This doesn't prove anything, though! To prove this, we need to look at areas. First notice that $$ODC \le ODA \le OBA $$, since each area contains the last are, plus another layer. Now, we can find the area of these three areas. The first area, ODC, is a triangle. This triangle has a base of $$ \,\ \cos(\theta)$$, and it has a height of $$ \,\ \sin(\theta)$$. Using the area formula for a triangle, we find
 * $$ODC=\frac{1}{2} \cos(\theta) \sin(\theta)$$
 * Now, we go on to the next area, ODA. It is a sector of a circle. The formula to find the area of the sector of a circle is
 * $$\frac {\theta r^2} {2}$$
 * This makes sense, since when you think that a complete circle would have two pi radians, the formula turns into the formula for a circle. In our unit circle, the area is simply $$\frac {\theta} {2}$$, since the radius is one.
 * The third area is also a triangle, with height of $$ \,\ \tan(\theta)$$, or $$\frac {\sin(\theta)}{\cos(\theta)}$$. With a base of one, its area turns into
 * $$\frac {1}{2} \frac {\sin(\theta)}{\cos(\theta)}$$
 * Now from the original expression of $$ODC \le ODA \le OBA $$, we have
 * $$\frac{1}{2} \cos(\theta) \sin(\theta)\le \frac {\theta}{2} \le \frac {1}{2} \frac {\sin(\theta)}{\cos(\theta)}$$
 * Multiply this whole inequality by two
 * $$ \cos(\theta) \sin(\theta) \le \theta \le \frac {\sin(\theta)}{\cos(\theta)}$$
 * Now divide the whole thing by $$ \,\ \sin(\theta)$$
 * $$ \cos(\theta) \le \frac {\theta} {\sin(\theta)} \le \frac {1}{\cos(\theta)}$$
 * Now we apply the limit!
 * $$ \lim_{\theta \to 0} \,\ \cos(\theta) \le \lim_{\theta \to 0} \,\ \frac {\theta} {\sin(\theta)} \le   \lim_{\theta \to 0 } \,\ \frac {1}{\cos(\theta)} \equiv 1 \le \lim_{\theta \to 0} \,\ \frac {\theta} {\sin(\theta)} \le \frac {1}{1}, \,\ or \,\ 1 $$

Further Study

 * Introduction to differentiation

Tools

 * Online Graphing Calculator

title: Introduction to Calculus/Limits


text:
Talk:Wuest type herringbone gear

Less wear? Really? Why? Better explanation needed...
Interesting concept, but why should it have better wear characteristics than a standard herringbone? Herringbone gears need to be used such that all but one of the gears in a meshed geartrain have end float on their shafts, so that the load on left and right sloping teeth are equal, and I don't see how you can do any better than that. Maybe if the tooth profile is not perfectly involute, the Wuest type will have any residual angular velocity errors at twice the frequency and half the amplitude of a standard herringbone, but there will be a residual alternating axial component too.

I feel that someone competent in the field needs to write a proper explanation before we can take it too seriously. Hopefully soon.

Tiger99 (talk) 21:57, 12 June 2019 (UTC)

title: Talk:Wuest type herringbone gear


text:
Calculus/Limits

Introducing Limits
Let's consider the following expression: $$\frac{n}{n+1}$$ As n gets larger and larger, the fraction gets closer and closer to 1. $$\left( \tfrac{1}{2}, \tfrac{2}{3}, \tfrac{3}{4}, \dots \tfrac{999}{1000} \right)$$

As n approaches infinity, the expression will evaluate to fractions where the difference between them and 1 becomes negligible. The expression itself approaches 1. As mathematicians would say, the limit of the expression as $$n \!$$ goes to infinity is 1, or in symbols: $$\lim_{n\rightarrow \infty}\frac{n}{n+1}=1$$.

An interesting sequence is $$S_{n}=\tfrac{1}{n}$$ As $$n \!$$ gets bigger (in symbols $$n\rightarrow\infty$$, we have smaller values of $$S \!$$, for
 * $$S_{1}=1 \!, S_{2}=\tfrac{1}{2}, S_{3}=\tfrac{1}{3}, S_{4}=\tfrac{1}{4}, S_{5}=\tfrac{1}{5}$$

and so on. Clearly, $$S \!$$ can't be smaller than zero (for if $$\tfrac{1}{n}<0$$ we have that $$n \!$$ is less than zero). Then we may say that $$\lim_{n\rightarrow \infty}S_{n}=0$$. Continuing with this sequence, we might want to study what happens when $$n \!$$ gets near to zero, and later what happens with negative values of $$n \!$$ going near to zero. Usually, the letter $$n \!$$ is reserved for integer values, so we are going to redefine our sequence as $$S_{n}=S(x)=\frac{1}{x}$$. If we take a sequence of values of $$x \!$$, say
 * $$x=1, x=\tfrac{1}{2}, x=\tfrac{1}{4}, x=\tfrac{1}{8}, x=\tfrac{1}{16} \!$$

We see that the respective values of $$S \!$$ grows indefinitely, for
 * $$S(1)=1 \!$$
 * $$S(2)=2 \!$$
 * $$S(4)=4 \!$$
 * $$S(8)=8 \!$$
 * $$S(16)=16 \!$$

In this case, we might say that the limit of $$\lim_{x\rightarrow 0^{+}}S(x)=\infty$$, in words, the limit of $$S \!$$ as $$x \!$$ goes to zero from right (as the sequence of values of $$x \!$$ goes to zero from the left in a graphic) diverge (or tends to infinity, or is unbounded, but we never say that it is infinity or equals infinity). Other case happens if we study sequences of values of $$x$$ such that every element of the sequence is lower than zero, the sequence is increasing but never exceeding zero. One example of such sequence is:
 * $$x=-1 \!$$, with $$S(-1)=-1 \!$$
 * $$x=-\frac{1}{2}$$, with $$S(-\frac{1}{2})=-2$$
 * $$x=-\frac{1}{4}$$, with $$S(-\frac{1}{4})=-4$$
 * $$x=-\frac{1}{8}$$, with $$S(-\frac{1}{8})=-8$$
 * $$x=-\frac{1}{16}$$, with $$S(-\frac{1}{16})=-16$$

The values of $$S(x)\! $$ decrease without bounds. The we say that $$\lim_{x\rightarrow 0^{-}}S(x)=-\infty$$, or that $$S(x)\!$$ tends to minus infinity when $$x$$ goes to zero from left.

Basic Limits
For some limits (if the function is continuous at and near the limit), the variable can be replaced with its value directly: $$\lim_{n\rightarrow x}f(n)=f(x)$$ For example, $$\lim_{x\rightarrow 5}\frac{x+10}{2x}=\frac{5+10}{2 \cdot 5}=\frac{3}{2}$$ and $$\lim_{m\rightarrow b}\frac{m+b}{m}=\frac{b+b}{b}=2$$ (with $$b$$ not equal to 0) Others are somewhat more complicated: $$\lim_{x\rightarrow \infty}\frac{5x+1}{x}=\lim_{x\rightarrow \infty}5+\frac{1}{x}=5+0=5$$ Note that in this limit, one may not immediately set $$x$$ equal to $$\infty$$ because this would result in the expression evaluating to $$\frac{\infty}{\infty}$$ which is an undefined expression. However, one may reduce the expression by separating the terms into separate fractions (in this case, $$\frac{5x}{x}$$ and $$\frac{1}{x}$$), which can be evaluated directly.

Right and Left Hand Limits
Sometimes, we want to calculate the limit of a function as a variable approaches a certain value from only one side; that is, from the left or right side. This is denoted, respectively, by $$\lim_{x\rightarrow_a-}$$ or $$\lim_{x\rightarrow_a+}$$. If the left-hand and right-hand limits do not both exist, or are not equal to each other, then the limit does not exist. The following limit does not exist: $$\lim_{x\rightarrow_0}\frac{1}{x}$$ It doesn't because the left and right handed limits are unequal. $$\lim_{x\rightarrow _0-} \frac{1}{x} = -\infty$$ $$\lim_{x\rightarrow _0+} \frac{1}{x} = \infty$$

Note that if the function is undefined at the point where we are trying to find the limit, it doesn't mean that the limit of the function at that point does not exist; for an example, let's evaluate $$g(x)=1/x^2$$ at $$x=0$$.

$$\lim_{x\rightarrow _0-} \frac{1}{x^2} = \infty = \lim_{x\rightarrow _0+} \frac{1}{x^2}$$

Therefore: $$\lim_{x\rightarrow _0} \frac{1}{x^2} = \infty$$

Some Formal Definitions and Properties
Until now, limits have been discussed informally but it shouldn't be all intuition, for we need to be sure of certain assertions. For an example, the limit
 * $$\lim_{n\rightarrow \infty}S_{n}=0$$

We have seen that the function decreases as $$n$$ increases, but how do we guarantee that there isn't a value $$y$$, say
 * $$y=0.000000000000000000000000000001$$

such that $$S$$ is never smaller than $$y$$? If there is such $$y$$, we might want to say that the limit is $$y$$, not zero, and we can't test every single possible value of $$y$$ (for there are infinite possibilities). We must then find a mathematical way of proving that there isn't such $$y$$, but for that we need to define formally what a limit is.

Right Limits, Left Limits, Limits and Continuity
Let $$f$$ be a real valued function. We say that
 * $$\lim_{x\rightarrow x_{0}^{-}}f(x)=c$$

if for every $$\epsilon>0$$ there is a $$\delta>0$$ such that, for every $$x'$$ between $$(x_{0}-\delta)$$ and $$x_{0}$$
 * $$|f(x')-c|<\epsilon$$

where $$|x|$$ is the absolute value of $$x$$.

This is the formal definition of convergence from the left. It means that for each possible error bigger than zero, we are able to find a interval such that for all $$x$$ in that interval, the distance between the value of the function and the constant $$c$$ is less than the error.

TODO: Graphics illustrating this.

In an analogous fashion, we say that
 * $$\lim_{x\rightarrow x_{0}^{+}}f(x)=c$$

if for every $$\epsilon>0$$ there is a $$\delta>0$$ such that, for every $$x'$$ between $$(x_{0})$$ and $$x_{0}+\delta$$, $$|f(x')-c|<\epsilon$$.

And to finish the necessary definitions,
 * $$\lim_{x\rightarrow x_{0}}f(x)=c$$

if
 * $$\lim_{x\rightarrow x_{0}^{-}}f(x)=c$$

and
 * $$\lim_{x\rightarrow x_{0}^{+}}f(x)=c$$.

Example:
 * $$\lim_{x\rightarrow 0}x^{2}=0$$

This is a assertion that must be proved. First, lets study the behavior of $$f(x)=x^{2}$$ near zero;
 * $$|x^{2}-0|<\epsilon\Rightarrow|x^{2}|<\epsilon\Rightarrow x^{2}<\epsilon\Rightarrow x<\sqrt{\epsilon}$$

where the arrow pointing right means implies. So, define the function
 * $$\delta(\epsilon)=\sqrt{\epsilon}$$

If $$|x-0|<\delta(\epsilon)$$, then $$|f(x)-0|<\epsilon$$. We have show how to find the delta in the definition of limit, showing that the limit of $$f$$ as x tends to zero is zero.

In fact, for any real number $$x_{0}$$,
 * $$\lim_{x\rightarrow x_{0}}x^{2}=x_{0}^{2}$$

Lets see how to construct a suitable function $$\delta(\epsilon)$$.
 * $$|x^{2}-x_{0}^{2}|=|(x-x_{0})(x+x_{0})|<(x+x_{0})^{2}$$, then $$(x+x_{0})^{2}<\epsilon\Rightarrow|x^{2}-x_{0}^{2}|<\epsilon$$

So,
 * $$(x+x_{0})^{2}<\epsilon\Rightarrow x+x_{0}<\sqrt{\epsilon}\Rightarrow \delta(\epsilon)=\sqrt{\epsilon}$$

implying that $$|x-x_{0}|<\delta(\epsilon)$$ makes $$|x^{2}-x_{0}^{2}|<\epsilon$$ for any $$x_{0}$$.

Functions with the property that
 * $$\lim_{x\rightarrow x_{0}}f(x)=f(x_{0})$$

are called continuous, and arise very naturally in the physical sciences; Beware that, against the intuition of most people, not every function is continuous.

Property of Limits
Property one: If $$\lim_{x\rightarrow x_{0}}f(x)=c$$, then
 * $$\lim_{x\rightarrow x_{0}}kf(x)=kc$$

for any constant $$k$$. Proof: Construct the function $$\delta(\epsilon)$$ for $$f$$. Then
 * $$|x-x_{0}|<\delta(\epsilon)\Rightarrow|f(x)-c|<\epsilon$$

So
 * $$|x-x_{0}|<\frac{1}{k}\delta(\epsilon)\Rightarrow|kf(x)-kc|=|k||f(x)-c|<\epsilon$$

Then the limit of $$kf$$ is $$kc$$, for the delta function of $$kf$$ is
 * $$\delta'(\epsilon)=\frac{1}{k}\delta(\epsilon)$$

QED.

TODO: Demonstrate main properties of limit (unicity, etc)

L'Hôpital's Rule
L'Hôpital's Rule is used when a limit approaches an indeterminate form. The two main indeterminate forms are $$\frac{0}{0}$$ and $$\frac{\infty}{\infty}$$. Other indeterminate forms can be algebrically manipulated, such as $$\infty-\infty$$.

L'Hôpital's Rule states if a limit of $$\frac{f(x)}{g(x)}$$ approaches an intederminate form as $$x$$ approaches $$a$$, then: $$\lim_{x\rightarrow a}\frac{f(x)}{g(x)} = \lim_{x\rightarrow a}\frac{f'(x)}{g'(x)}$$

Example: $$\lim_{x\rightarrow 0}\frac{\sin(x)}{x}$$ Both the numerator and the denominator approach zero as $$x$$ approaches zero, therefore the limit is in indeterminate form, so l'Hôpital's rule can be applied to this limit. (note: you can also use the Sandwich Theorem.) $$\lim_{x\rightarrow 0}\frac{\sin(x)}{x} = \lim_{x\rightarrow 0}\frac{\sin'(x)}{x'} = \lim_{x\rightarrow 0}\frac{\cos(x)}{1}$$ Now the limit is in a usable form, which equals 1.

If the limit resulting from applying l'Hôpital's Rule is still one of the two mentioned indeterminates, we may apply the rule again (to the limit obtained), and again and again until a usable form is encountered.

Where does it come from
To obtain l'Hôpital's Rule for a limit of $$\frac{f(x)}{g(x)}$$ which approaches $$\frac{0}{0}$$ as $$x$$ approaches $$a$$, we simply decompose both $$f(x)$$ and $$g(x)$$ in terms of their Taylor expansion (centered around $$a$$). The independent terms of both expansions must be $$0$$ (because both $$f(x)$$ and $$g(x)$$ approached $$0$$), so if we divide both the $$f(x)$$ and $$g(x)$$ by $$(x-a)$$ (or, equivalently, find their derivatives), our limit will stop being indeterminate.

It could be the case that the Taylor expansions of both the numerator and the denominator have a $$0$$ as coefficient of the $$(x-a)$$ term, thus yielding an indeterminate. This is the same case mentioned above where the trick was to repeat the process until a suitable limit was found.

The case of a limit which approaches $$\frac{\infty}{\infty}$$ can be transformed to the case above by exchanging $$\frac{f(x)}{g(x)}$$ with $$\frac{1/g(x)}{1/f(x)}$$, which obviously approaches $$\frac{0}{0}$$.

Calculus/Limits/Exercises

title: Calculus/Limits