Question

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

Answer 1

Answer: 3

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