limit

Intuitively, limit is what you can expect the value of a function to be (or approach) without knowing the actual value (say by looking at a graph without that point). A limit exists if and only if ${\lim }_{x\to c^{-}}f(x)={\lim }_{x\to c^{+}}f(x)$

Formally, if for every $ϵ>0$, there exists $\delta >0$ such that for all $x\in A$, $|x-a|<\delta$ and $|f(x)-L|<ϵ$, then we say $f$ converges to L as $x\to a$.

When calculating a limit at a endpoint, only a one-sided limit is sufficient for the limit to be defined (i.e. only the graph within the domain needs to be taken into account).

Notation

${\lim }_{x\to c}f(x)$ is read as “the limit of $f(x)$ as x approaches c” ${\lim }_{x\to c^{-}}f(x)$ is the limit from the left (negative) side, a.k.a. the left-handed limit ${\lim }_{x\to c^{+}}f(x)$ is the limit from the right (positive) side, a.k.a. the right-handed limit