Question

$\underset{x\to 0}{\lim }\frac{si{n}^{2}x}{\sqrt{2}-\sqrt{1+cosx}}$ equal to $a\sqrt{2}$ , then $a$ is

Answer 1

Answer: 4

$\underset{x\to 0}{\lim }\frac{si{n}^{2}x}{\sqrt{2}-\sqrt{1+cosx}}$
$=\underset{x\to 0}{\lim }\frac{si{n}^{2}x}{\sqrt{2}-\sqrt{2co{s}^2\frac{x}{2}}}[\frac{0}{0}]$
$=\underset{x\to 0}{\lim }\frac{si{n}^{2}x}{\sqrt{2}[1-cos\frac{x}{2}]}=\underset{x\to 0}{\lim }\frac{si{n}^{2}x}{2\sqrt{2}si{n}^2\frac{x}{4}}$
$=\underset{x\to 0}{\lim }\frac{{\left(\frac{sinx}{x}\right)}^2.16}{2\sqrt{2}{\left(\frac{sin\frac{x}{4}}{\frac{x}{4}}\right)}^2}$
$=\frac{16}{2\sqrt{2}}=4\sqrt{2}$
$\Rightarrow a\sqrt{2}=4\sqrt{2}$
$\Rightarrow a=4$