Question

Prove that the function
$\rm f(x)=\begin{cases}\dfrac{x}{|x|}&\text{if}\:x\neq 0 \\ \\ 1&\text{if}\:x=0\end {cases}$
is not differentiable at x=0​

Answer 1

$\rm f|x|=\begin{cases}{x}&\text{if}\:x\ge 0 \\ \\ -x&\text{if}\:x<0\end {cases}$

$\sf{\huge{LHL :-}}$

$\qquad\quad\sf{\large{\lim_{x\to 0^-=f(x)}}}$

$\qquad\quad\sf{\large{\lim_{x\to 0^-=\frac{x}{-x}=\lim_{x\to 0^-=-(1)}}}}$

$\sf{\huge{RHL :-}}$

$\qquad\quad\sf{\large{\lim_{x\to 0^+=f(x)}}}$

$\qquad\quad\sf{\large{\lim_{x\to 0^+=\frac{x}{x}=\lim_{x\to 0^+=1}}}}$

Answer 2

Answer:

your answer is in the attachment