Question:
If the function $f(x)=\frac{\sin 10 x}{x}, x \neq 0$ is continuous at $x=0$, find $f(0)$.
Solution:
Given: $f(x)=\frac{\sin 10 x}{x}, x \neq 0$ is continuous at $x=0$.
$\lim _{x \rightarrow 0} f(x)=f(0)$
$\Rightarrow \lim _{x \rightarrow 0} \frac{\sin 10 x}{x}=f(0)$
$\Rightarrow \lim _{x \rightarrow 0} \frac{10 \sin 10 x}{10 x}=f(0)$
$\Rightarrow 10 \lim _{x \rightarrow 0} \frac{\sin 10 x}{10 x}=f(0)$
$\Rightarrow f(0)=10$