Question

Evaluate the left and right hand limits of the function $f\left(x\right)=\{\begin{array}{cc}\frac{|x-4|}{x-4},& x\ne 4\\ 0,& x=4\end{array}$ at $x=4$

• A. Both LHL and RHL are 0.
• B. LHL is -1 and RHL is +1.
• C. LHL is +1 and RHL is -1.
• D. Both LHL and RHL are 4.

Answer 1

Answer: (B)

LHL is -1 and RHL is +1.

LHL of $f\left(x\right)$ at $x=4$ is
$\underset{x\to 4^{-}}{lim}f\left(x\right)=\underset{h\to 0}{lim}f(4-h)$
$=\underset{h\to 0}{lim}\frac{|4-h-4|}{4-h-4}$
$=\underset{h\to 0}{lim}\frac{|-h|}{-h}$
$=\underset{h\to 0}{lim}\frac{h}{-h}$
$=\underset{h\to 0}{lim}-1=-1$
RHL of $f\left(x\right)$ at $x=4$ is
$\underset{x\to 4^{+}}{lim}f\left(x\right)=\underset{h\to 0}{lim}f(4+h)$
$=\underset{h\to 0}{lim}\frac{|4+h-4|}{4+h-4}$
$=\underset{h\to 0}{lim}\frac{|h|}{h}$
$=\underset{h\to 0}{lim}\frac{h}{h}$
$=\underset{h\to 0}{lim}1=1$