TITLE: Understanding of the formal and intuitive definition of a limit
QUESTION [8 upvotes]: The intuitive definition for $\lim\limits_{x \to a} f(x) = L$ is the value of $f ( x )$ can be made arbitrarily close to $L$ by making $x$ sufficiently close,  but not equal to, $a$ .
I can easily understand this ,but for the (ε, δ)-definition of limit:For every real ε > 0, there exists a real δ > 0 such that for all real x, 0 < | x − a | < δ implies | f(x) − L | < ε. Oh, god, I cannot understand it completely.
As it is the formal definition of limit, I think it should be precise but somewhat should include the mean of the above intuitive definition, so as for "$f ( x )$ can be made arbitrarily close to $L$" in the intuitive definition correspond to “For every real ε > 0,| f(x) − L | < ε” in the formal definition, it's fine! but does  “making $x$ sufficiently close,  but not equal to, $a$ .” correspond to “there exists a real δ > 0 such that for all real x, 0 < | x − a | < δ” ? This is the point I cannot understand, because I am not sure if  “there exists a real δ > 0 such that for all real x, 0 < | x − a | < δ”  shows "x close enough, but not equal, to $a$".
The other question is: does the biggest δ also get smaller as ε is getting smaller? Why? (Exclude the case when f(x) is a constant function.)
Why do we need the formal definition of a limit? Does the intuitive definition have some flaw?
P.S. Thank you everyone, but I must declare I only have some basic knowledge of limit, I only started to learn calculus a few days ago.

REPLY [2 votes]: First, notice that

$f(x)$ can be made arbitrarily close to $L$ by choosing $x$ close enough to $a$

is equivalent to

No matter how close you want $f(x)$ and $L$ to be, you can limit the distance between $x$ and $a$ such that $f(x)$ and $L$ will be as close as you wanted.

Let's give names ($\varepsilon$ and $\delta$) to the two "close" distances:

For any maximum distance $\varepsilon > 0$ between $f(x)$ and $L$, there is a maximum distance $\delta > 0$ between $x$ and $a$, such that whenever $x$ and $a$ are closer than $\delta$, $f(x)$ and $L$ are closer than $\varepsilon$.

Removing redundant words, we get:

For any $\varepsilon > 0$, there is a $\delta > 0$, such that whenever $x$ and $a$ are closer than $\delta$, $f(x)$ and $L$ are closer than $\varepsilon$.

And using $|a - b|$ instead of "distance between $a$ and $b$", we get:

For any $\varepsilon > 0$, there is a $\delta > 0$, such that whenever $|x - a| < \delta$, $|f(x) - L| < \varepsilon$.