Question

If the value of $\displaystyle\lim _{x \rightarrow 0}(2-\cos x \sqrt{\cos 2 x})^{\left(\frac{x+2}{x^{2}}\right)}$ is equal to $e^{a}$, then a is equal to_______

Answer 1

$\displaystyle\lim _{x \rightarrow 0}(2-\cos x \sqrt{\cos 2 x})^{\left(\frac{x+2}{x^{2}}\right)}$
form: $1^{\infty}$
$=e^{\displaystyle\lim _{x \rightarrow 0}}\left(\frac{1-\cos x \sqrt{\cos 2 x}}{x^{2}}\right) \times(x+2)$
Now $\displaystyle\lim _{x \rightarrow 0} \frac{1-\cos x \sqrt{\cos 2 x}}{x^{2}}$
$=\displaystyle\lim _{x \rightarrow 0} \frac{\sin x \sqrt{\cos 2 x}-\cos x \times \frac{1}{2 \sqrt{\cos 2 x}} \times(-2 \sin 2 x)}{2 x}$
(by L' Hospital Rule)
$\displaystyle\lim _{x \rightarrow 0} t \frac{\sin x \cos 2 x+\sin 2 x \cdot \cos x}{2 x}$
$=\frac{1}{2}+1=\frac{3}{2}$
So, $e^{\displaystyle\lim_{x\rightarrow 0} t\left( \frac{1-\cos x \sqrt{\cos 2 x}}{x^{2}}\right)(x+2)}$
$=e^{\frac{3}{2} \times 2}=e^{3} $
$\Rightarrow a=3$