NIMCET 2018 Question Paper with solutions

We have.
f (x) =
Given function f is differentiable at all points except possible at x = 0.
∴ L f ' (0) = $\underset{\mathit{x}\to 0^{-}}{\lim }$ $\frac{\mathit{f}\left(\mathit{x}\right)-\mathit{f}\left(0\right)}{\mathit{x}-0}$
= $\underset{\mathit{x}\to 0^{-}}{\lim }$ $\frac{\frac{\mathit{x}}{1-\mathit{x}}}{\mathit{x}}$ = $\underset{\mathit{x}\to 0^{-}}{\lim }$ $\frac{1}{1-\mathit{x}}$ = 1
and R f' (0) = $\underset{\mathit{x}\to 0^{+}}{\lim }$ $\frac{\mathit{f}\left(\mathit{x}\right)-\mathit{f}\left(0\right)}{\mathit{x}-0}$
= $\underset{\mathit{x}\to 0^{+}}{\lim }$ $\frac{\left(\frac{\mathit{x}}{1+\mathit{x}}\right)}{\mathit{x}}$ = $\underset{\mathit{x}\to 0^{+}}{\lim }$ $\frac{1}{1+\mathit{x}}$ = 1
L f ' (0) = R f ' (0) = 1 ,
we get
f is differentiable at x = 0,Thus, f is differentiable for all x ∊ (- ∞ , ∞)