# What does limit does not exist mean?

Apr 13, 2018

For the definition, please see below.

#### Explanation:

${\lim}_{x \rightarrow a} f \left(x\right)$ does not exist
The idea is that there is no number that $f \left(x\right)$ gets arbitrarily close to for $x$ sufficiently close to $a$.

For a function $f$ defined in some open interval that contains $a$, except possibly at $a$,

${\lim}_{x \rightarrow a} f \left(x\right)$ does not exist if and only if

there is no number $L$ such that for every $\epsilon > 0$, there is a $\delta > 0$ with: for all $x$, if $0 < \left\mid x - a \right\mid < \delta$, then $\left\mid f \left(x\right) - L \right\mid < \epsilon$

equivalently

for every $L$ there is some $\epsilon > 0$ such that for every $\delta > 0$ there is some $x$ with $0 < \left\mid x - a \right\mid < \delta$, and $\left\mid f \left(x\right) - L \right\mid \ge \epsilon$