TITLE: Epsilon-Delta Confusion
QUESTION [7 upvotes]: I don't understand the epsilon delta definition of a limit 
"According to the formal definition above, a limit statement is correct if and only if confining  x to d units of c will inevitably confine f(x) to epsilon units of L."
So, if we can confine x to infinitely small delta units of c, we can confine f(x) to infitely small epsilon units of L.  Like, constricting f(x) to L ?
Is that a right way of explaining what the general idea of the epsilon-delta method is ?

REPLY [2 votes]: Let $y \in \mathbb{R}$ and $f$ a function whose domain is defined for points near $y$. We say $\lim_{x \rightarrow y} f(x)=L \in \mathbb{R}$ iff for each $\epsilon >0$ there exists $\delta >0$ such that if $x \in \mathbb{R}$ and $0 < |x-y| < \delta$ then $|f(x)-L|<\epsilon$.
Notice the $0<|x-y|$ means we do not consider the value $x=y$ for the limiting process. Now, it is often the case that $L=f(y)$ and the limit is calculated by mere evaluation of the function $f$; this is the case of a function which is continuous at $y$. In fact, this is the definition of continuity at $y$ for $f$. We say $f$ is continuous at $x=y$ iff $\lim_{x \rightarrow y}f(x) = f(y)$. Unless $y$ is an endpoint of the domain or an isolated point then some fine print applies, but let us focus on the essential point here.
One important example (there are others, but certainly this class of examples is unavoidable in the study of calculus!) of the exclusion of $x=y$ in the limit is found in difference quotients: if the limit 
$$  \lim_{x \rightarrow y} \frac{f(x)-f(y)}{x-y} $$
exists then we call it $f'(y)$ the derivative of $f$ at $y$. It is never reasonable to just plug $x=y$ into the difference quotient because it necessitates division by zero. So, the $0$ in the definition "$0 < |x-y|$" is of critical importance because it is the subtle point which separates taking a limit from just doing algebra. Although, I will admit, the majority of the limits we encounter in first semester calculus are just that; algebra. But, conceptually, the exclusion of the limit point from the limit is not optional in my view. It is the point.
(of course, again, for continuous expressions there is no difference, but continuity cannot be expected in many cases)