Discuss the differentiability of `f(x)={x e-(1/(|x|)+1/x),x!=0x ,x=0a tx=0`

Explanation

Given function,
`f(x)={{:(xe^-(1/(|x|)+1/x),x!=0),(0,if x=0):}`
The value of `f(x)` at `x=h` where` h→0` is `(he)^(-1/(2h))=0`
Similarly, `f(−h)=−he^0=0`
So, the function is continuous at `x=0`
Now,
`f'(0^+)=lim_(h->0)((he)^(-1/(2h)))/h=e^(-1/(2h))=0`
`f'(0^+)=lim_(h->0)(-he^0)/-h=e^0=1`
Hence the function isn't differentiable at` x=0`.