Question

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

Answer 1

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

We have
$$\begin{gathered} \underset{x\to 0}{\lim }f\left(x\right)=\underset{x\to 0}{\lim }\frac{\sin x^2}{x} \\ =\underset{x\to 0}{\lim }\frac{x\sin x^2}{x^2} \\ =\underset{x\to 0}{\lim }\frac{\sin x^2}{x^2}\underset{x\to 0}{\lim }x \\ =1\times 0 \\ =0 \\ =f\left(0\right) \end{gathered}$$

$\underset{x\to 0}{\therefore \lim }f\left(x\right)=f\left(0\right)$

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