Answer
$f(x)$ is not differentiable at $P$.
Work Step by Step
Consider $f(x)=2x$.
The right-hand derivative of $f(x)$ at $P(1,2)$ is given as:
$\lim\limits_{h\to0^+}\dfrac{f(h+1)-f(1)}{h}=\lim\limits_{h\to0^+}\dfrac{2h+2-2}{h}=\lim\limits_{h\to0^+}\dfrac{2h}{h}=2$ ...(1)
The left-hand derivative of $f(x)$ at $P(1,2)$ is given as:
we have $f(x)=2$
Thus$\lim\limits_{h\to0^-}\dfrac{f(h+1)-f(1)}{h}=\lim\limits_{h\to0^-}\dfrac{2-2}{h}=0$ ...(2)
From the above equations (1) and (2), we conclude that the left-hand derivative is not equal to the right-hand derivative. This means that $f(x)$ is not differentiable at $P$.
