Let us shift the graph of
f(x) in the direction of the negative
x−axis by
4 units
& let
g(x)=f(x+4)∵g(x) at
x=0 will have the same behaviour as of
f(x) at
x=4g\left(x\right) = \left\{sinx.tan−1(x1)0$\begin{array}{ll}x \ne 0 \\ x = 0\end{array}\right)
Continuity at
x=0,
limx→0−g(x)=limx→0−sinx.tan−1(x1)=0×(−π/2)=0limx→0+g(x)=limx→0+sinx.tan−1(x1)=0×(π/2)=0∴limx→0+g(x)=g(0)∴g(x) is continuous at
x=0⇒f(x) is continuous at
x=4Differentiability at
x=0g′(0−)=limh→0+−hg(0−h)−g(0)=limh→0+−hsin(−h)tan−1(h−1)−0⇒limh→0+−(hsinh)tan−1(h1)=−1.(2π)=2−πg′(0+)=limh→0+hg(0+h)−g(0)=limh→0+hsin h.tan−1h1−0⇒limh→0+(hsinh)tan−1(h1)=(2π)Clearly,
g′(0−)=g′(0+)∴g(x) is non-differentiable at
x=0⇒f(x) is non-differentiable at $x = 4
