TITLE: What concepts were most difficult for you to understand in Calculus?
QUESTION [13 upvotes]: I'm developing some instructional material for a Calculus 1 class and I wanted to know from experience for yourself, tutoring others, and/or helping people on this site where is the most difficulty in Calculus? 
If you had any good methods of helping people that would be very helpful.

REPLY [2 votes]: The logic behind the $\mathbb {\epsilon}$ - $\mathbb { \delta}$ definition.

This might have been mentioned in an answer above but since you are a teacher (or are instructing one) I think I must express some frustration I have had with your standard freshman year Calculus course. 
My teacher - and many others I've heard - take the $\epsilon$ - $\delta$ definition of limits for granted. They just get away with repeating the statement "For any given $\epsilon \gt 0 $ $ \;\;\; \exists \delta \gt 0$ such that $|f(x) - L| \lt \epsilon $ whenever $|x - a| \lt \delta$". The students in my country (Sri Lanka) get entrance to university based on a very competitive final paper in school. Almost all of them are very capable of strong and mind-wrecking computations. But almost all of them have difficulty understanding limits in their first semester, as a result hate calculus and then despise pure mathematics in general. And I blame the teaching for this trend. The tutor fails to instill on the student the logic behind the definition. Not many know the fact that you are required to prove the existence of $\delta$ and not just an implication $|x - a| \lt \delta \implies |f(x) - L| \lt \epsilon $. It has been a couple of months since I joined this site and I repeatedly come across questions posted with similar dilemmas and all of them have flimsy logical foundation. And that is the issue. 
So my suggestion here is a better introduction to the logic behind the introductory calculus courses. One professor I know starts off by asking students to negate statements like "Every girl is pretty" and "Some parents are kind" which I think is an excellent approach. 
I found it very difficult to grasp the sense behind the limit statement and took me long hours and several books to get a hold of it. And I think this can and should be avoided. Yes a student should work out problems on his own and do lot of work on his own. But the foundation should be laid. Solidly. And I think the teacher is responsible for that.