Question

The function $f$ defined as $f\left(x\right)=\{\begin{array}{c}\frac{\sin x^2}{x}\\ 0,x=0\end{array},x\ne 0$

• A. is continuous and differentiable at $x=0$
• B. is neither continuous nor differentiable at $x=0$
• C. is continuous but not differentiable at $x=0$
• D. None of these

Answer 1

The correct option is A is continuous and differentiable at $x=0$

$f\left(x\right)=\{\begin{array}{c}\frac{\sin x^2}{x},x\ne 0\\ 0,x=0\end{array},$
$f\left(x\right)=\{\begin{array}{c}x.\frac{\sin x^2}{x^2},x\ne 0\\ 0,x=0\end{array},$
L.H.L $=0\left(1\right)=1=$ R.H.L $=f\left(0\right)\Rightarrow f\left(x\right)$ is continuous at $x=0$
$f^{\prime }\left(x\right)=\{\begin{array}{c}\frac{2x^2\cos x^2-\sin x^2}{x^2},x\ne 0\\ 0,x=0\end{array},$
L.H.D $=0-0=0=$ R.H.D. Hence $f\left(x\right)$ is differentiable as well at $x=0$