Home -> Solved problems -> Calculate limit when it exists

## Calculate the following limit, if it exists

### Solution

Writing the definition of the absolute value of $$x$$
$|x|={\begin{cases}-x\;\;,\;\;\;\;\;{\text{if }}x< 0\\x\;\;\;\;\;,\;\;\;\;\;{\text{if }}x\geq0\end{cases}}$
$\Rightarrow\lim_{x \rightarrow 0^{+}}\frac{|x|}{x}=\lim_{x \rightarrow 0^{+}}\frac{x}{x}$
$=\lim_{x \rightarrow 0^{+}}1$
$=1$
$\Rightarrow\lim_{x \rightarrow 0^{-}}\frac{|x|}{x}=\lim_{x \rightarrow 0^{-}}\frac{-x}{x}$
$=\lim_{x \rightarrow 0^{-}}-1$
$=-1$
$\Rightarrow\lim_{x \rightarrow 0^{+}}\frac{|x|}{x}\neq\lim_{x \rightarrow 0^{-}}\frac{|x|}{x}$
Therefore,
$\Rightarrow\lim_{x \rightarrow 0}\frac{|x|}{x}\;\;\;\;\;\;\;Does\;not\;exist$
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