Question

If the function $f\left(x\right)=\frac{\sin 10x}{x},x\ne 0$ is continuous at x = 0, find f (0).

Answer 1

Given: $f\left(x\right)=\frac{\sin 10x}{x},x\ne 0$ is continuous at $x=0$.
$\underset{x\to 0}{\lim }f\left(x\right)=f\left(0\right)$

$\Rightarrow \underset{x\to 0}{\lim }\frac{\sin 10x}{x}=f\left(0\right)$

$$\begin{gathered} \Rightarrow \underset{x\to 0}{\lim }\frac{10\sin 10x}{10x}=f\left(0\right) \\ \Rightarrow 10\underset{x\to 0}{\lim }\frac{\sin 10x}{10x}=f\left(0\right) \\ \Rightarrow f\left(0\right)=10 \end{gathered}$$