Intro To Limits

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When limits don't exist

For $\displaystyle\lim_{x \to a} f(x)$ to exist and equal $L$, we need $f(x)$ to be approximately $L$ on both sides of $x=a$.

$$\lim_{x \rightarrow a}f(x)=L \qquad \Longleftrightarrow\qquad \lim_{x \rightarrow a^-}f(x)=L \text{ and } \lim_{x \rightarrow a^+}f(x)=L.$$

If the two one-sided limits are different, or if one (or both) of them fail to exist, then the overall limit doesn't exist.

Note that the value of $f(a)$ doesn't enter into this, and it doesn't matter whether $f(a)$ is defined or not defined. Recall that when evaluating a limit of $f$ as $x\to a$ we only care about what happens when $x$ is near $a$ (when $x$ is slightly less than $a$, or slightly greater than $a$), not what happens when $x$ is equal to $a$.

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