The Limit of a Function

This page is intended to be a part of the Real Analysis section of Math Online. Similar topics can also be found in the Calculus section of the site.

 Definition: Let $f : A \to \mathbb{R}$ be a function and let $c$ be a cluster point of $A$. Then we say $L \in \mathbb{R}$ is the limit of $f$ at $c$ written $\lim_{x \to c} f(x) = L$ if $\forall \epsilon > 0$ there exists a $\delta > 0$ such that $\forall x \in A$ with $0 < \mid x - c \mid < \delta$ we have that $\mid f(x) - L \mid < \epsilon$. Equivalently this definition can be rephrased as $\lim_{x \to c} f(x) = L$ if $\forall \epsilon > 0 \: \exists \delta > 0$ such that if $x \neq c$ and $x \in V_{\delta} (c) \cap A$ then $f(x) \in V_{\epsilon} (L)$.
We should make mention that the definition of a $L$ being the limit of a function $f$ at the point $x = c$ does not require that $c$ be in the domain, $A$ of $f$. Instead, the definition of the limit of a function is only concerned that $c$ is a cluster point of the domain $A$, that is for $\delta > 0$, then for any delta-neighbourhood around $a$, there exists at least one point in $A$ that differs from $c$. This ensures that for very small $\delta$ there are points in the domain of $f$ for which the limit can be determined.
The following three pages regard the limit of a function. The first page regards the uniqueness of the limit of a function at the cluster point $c$ of $A$, that is if $\lim_{x \to c} f(x) = L$ and $\lim_{x \to c} f(x) = M$ then $L = M$. The second page regards criterion in terms of sequences for which a function has limit $L$ at $c$, while the third page regards criteria in terms of sequences for which a function does NOT have limit $L$ at $c$.