Show that _ x 4 |x-4| x-4 does not exist. - Vidyadip

Question

Show that $\lim _{x \rightarrow 4} \frac{|x-4|}{x-4}$ does not exist.

✓

Answer

Let $x-4=n \Rightarrow$ As $x \rightarrow 4, n \rightarrow 0$
$
\begin{aligned}
\therefore \text { L.H.L } & =\lim _{n \rightarrow 0-} \frac{|n|}{n}=\lim _{n \rightarrow 0-} \frac{-n}{n} \\
& =\lim _{n \rightarrow 0-}(-1)=-1
\end{aligned}
$
$\begin{aligned} \text { R.H.L } & =\lim _{n \rightarrow 0+} \frac{|n|}{n}=\lim _{n \rightarrow 0+} \frac{n}{n} \\ & =\lim _{n \rightarrow 0+}(1)=1\end{aligned}$
Since L.H.L $\neq$ R.H.L
$\lim _{x \rightarrow 4} \frac{|x-4|}{x-4}$ does not exist.

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