Definition of Limits

An initial definition provides an intuitive understanding of limits.

Definition. Suppose $f(x)$ is defined when $x$ is near the number $a$ (This means that $f$ is defined around $a$, except possibly at $a$ itself.). Then we write \(\lim_{x\to a}f(x)=L\) and say “the limit of $f(x)$, as $x$ approaches $a$, equals $L$” if we can make the values of $f(x)$ arbitrarily close to $L$ by restricting $x$ to be sufficiently close to $a$ but not equal to $a$.

When we consider both the left and right limits, we define a limit $\displaystyle\lim_{x\to a} f(x)$ to exist if and only if both the left and right limits exist and satisfy the condition: \(\lim_{x\to a^{-}}f(x) = \lim_{x\to a^+}f(x)\)

The the precise definition of limits uses the language of $\varepsilon$ and $\delta$.