Question

Find $\underset{x\to 5}{lim}f\left(x\right),$ where $f\left(x\right)=|x|-5$

Answer 1

Given : $f\left(x\right)=|x|-5$
Limit of the given function at $x=5$ exists if $\underset{x\to 5^{-}}{lim}f\left(x\right)=\underset{x\to 5^{+}}{lim}f\left(x\right)$

$L.H.L.=\underset{x\to 5^{-}}{lim}f\left(x\right)=\underset{h\to 0}{lim}f(5-h)$
$\Rightarrow L.H.L.=\underset{h\to 0}{lim}|5-h|-5$
$\Rightarrow L.H.L.=\underset{h\to 0}{lim}(5-h)-5$
$\Rightarrow L.H.L.=\underset{h\to 0}{lim}-h=0$

$R.H.L.=\underset{x\to 5^{+}}{lim}f\left(x\right)=\underset{h\to 0}{lim}f(5+h)$
$\Rightarrow R.H.L.=\underset{h\to 0}{lim}|5+h|-5$
$\Rightarrow R.H.L.=\underset{h\to 0}{lim}(5+h)-5$
$\Rightarrow R.H.L.=\underset{h\to 0}{lim}h=0$

$L.H.L=R.H.L=0$
Thus,$\underset{x\to 5}{lim}f\left(x\right)=0$