Question

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

• A. continuous and derivable at $x=0$
• B. neither continuous nor derivable at $x=0$
• C. continuous but not derivable at $x=0$
• D. continuous and derivable at $x=1$

Answer 1

Answer: (A)

continuous and derivable at $x=0$

Let multiply and divide by $x$

$f\left(x\right)=\underset{x\to 0}{lim}\frac{x\sin x^2}{x^2}=0$

So $f\left(x\right)$ to continous

$f^{\prime }\left(x\right)=\frac{2x\cos x^2}{x}-\frac{\sin \left(x^2\right)}{x^2}$

$\underset{x\to 0}{lim}{f}^{\prime }\left(x\right)=\underset{x\to 0}{lim}\cos x^2-\frac{\sin x^2}{x^2}$

$=1-1=0$

So $f^{\prime }\left(x\right)$ is also continous

& $f\left(x\right)$ is derivable