Question

For what value of k is the function
$f\left(x\right)=\left\{\begin{array}{ll}\frac{\sin 2x}{x},& x\ne 0\\ k,& x=0\end{array}\right.$
continuous at x = 0?

Answer 1

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

If f(x) is continuous at x = 0, then

​$\underset{x\to 0}{\lim }f\left(x\right)=f\left(0\right)$
$\Rightarrow \underset{x\to 0}{\lim }\frac{\sin 2x}{x}=k$
$\Rightarrow \underset{x\to 0}{\lim }\frac{2\sin 2x}{2x}=k$
$\Rightarrow 2\underset{x\to 0}{\lim }\frac{\sin 2x}{2x}=k$
$\Rightarrow 2\times 1=k$
$\Rightarrow k=2$