Determine whether


Determine whether $f(x)=\left\{\begin{array}{cl}\frac{\sin x^{2}}{x}, & x \neq 0 \\ 0 & , x=0\end{array}\right.$ is continuous at $x=0$ or not.


Given: $f(x)=\left\{\begin{array}{l}\frac{\sin x^{2}}{x}, x \neq 0 \\ 0, x=0\end{array}\right.$

We have

$\lim _{x \rightarrow 0} f(x)=\lim _{x \rightarrow 0} \frac{\sin x^{2}}{x}$

$=\lim _{x \rightarrow 0} \frac{x \sin x^{2}}{x^{2}}$

$=\lim _{x \rightarrow 0} \frac{\sin x^{2}}{x^{2}} \lim _{x \rightarrow 0} x$

$=1 \times 0$



$\therefore \lim _{x \rightarrow 0} f(x)=f(0)$

Hence, $f(x)$ is continuous at $x=0$.

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