Question

If $f\left(x\right)=\{\begin{array}{cc}{x}^2\sin \frac{1}{x}& x\ne 0\\ 0& x=0\end{array}$ then R.H.D. is

• A. $1$
• B. $-1$
• C. $0$
• D. None

Answer 1

Answer: (C)

$0$

Given: $f\left(x\right)=\{\begin{array}{c}{x}^2\sin \left(\frac{1}{x}\right)x\ne 0\\ 0x=0\end{array}$

To Find: R.H.D

Sol:$f^{\prime }\left(x\right)=\underset{h\to 0}{lim}\frac{f(x+h)-f\left(x\right)}{x+h-x}$

$\Rightarrow f^{\prime }\left(0\right)$=$\underset{h\to 0}{lim}$ $\frac{f\left(h\right)-f\left(0\right)}{h}$

$\underset{h\to 0^{+}}{lim}$ $\frac{f\left(h\right)}{h}$=$\underset{h\to 0^{+}}{lim}$ $\frac{h^2\sin \frac{1}{h}}{h}$=$\underset{h\to 0^{+}}{lim}$ $h.\sin \left(\frac{1}{h}\right)$

=$0$

Hence, correct answer is zero.