Question

Evaluate the left hand limit of the function $f\left(x\right)=\{\begin{array}{c}\frac{|x-4|}{x-4},x\ne 4\\ 0,x=4\end{array}$

Answer 1

$f\left(x\right)=\{\begin{array}{c}\frac{|x-4|}{x-4},x\ne 4\\ 0,x=4\end{array}$

Evaluating left hand limit at x = 4

$\underset{x\to h-4}{lim}=\left\{\frac{|h-4-4|}{h-4-4}\right\}=\frac{|h-8|}{h-8}$

$h\to 0$

$=\left|\frac{|0-8|}{0-8}\right|h=0\Rightarrow \frac{|-8|}{-8}=\frac{8}{-8}=-1$