Question

Find the value of $f\left(0\right)$ for which $f\left(x\right)=\frac{64(\sqrt{x+4}-2)}{\sin 2x}$ is continuous.

Answer 1

$f\left(0\right)=64\cdot \underset{x\to 0}{lim}\frac{\sqrt{x+4}-2}{sin2x}$
$=64\cdot \underset{x\to 0}{lim}\frac{\sqrt{x+4}-2}{sin2x}\cdot \frac{\sqrt{x+4}+2}{\sqrt{x+4}+2}$
$=64\cdot \frac{1}{2}\cdot \underset{x\to 0}{lim}\frac{2x}{sin2x}\cdot \frac{1}{\sqrt{x+4}+2}$
$=64\cdot \frac{1}{2}\cdot 1\cdot \frac{1}{\sqrt{4}+2}=8$