_ x a x-a |x-a| = - Vidyadip

MCQ

$\lim_\limits{\text{x} \rightarrow \text{a}}\frac{\text{x}-\text{a}}{|\text{x}-\text{a}|}=$

A
$0$
B
$1$
C
$-1$
✓
does not exist

✓

Answer

Correct option: D.

does not exist

Using,
$\lim_\limits{\text{x} \rightarrow 0}|\text{x}|=-\text{x}$
$\lim_\limits{\text{x} \rightarrow 0}|\text{x}|=+\text{x}$
we get $\lim_\limits{\text{x} \rightarrow \text{a}}-\frac{\text{x}-\text{a}}{-(\text{x}-\text{a})}=-1$
$\lim_\limits{\text{x} \rightarrow \text{a}}+\frac{\text{x}-\text{a}}{-(\text{x}-\text{a})}=-1$
Since, $\text{LHL}$ is not equal to $\text{RHL}$,
hence the limit does not exist.

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