Question

Find $\underset{x\to \infty }{lim}\sqrt{\frac{x-\sin x}{x+{\cos }^{2}x}}$

Answer 1

Given equation is

$=\underset{x\to \infty }{lim}\sqrt{\frac{x-\sin x}{x+{\cos }^{2}x}}$

Let,

$x=\frac{1}{t}$ as $x\to \infty ,t\to 0$

$=\underset{t\to 0}{lim}\sqrt{\frac{\frac{1}{t}-\sin \frac{1}{t}}{\frac{1}{t}+{\cos }^2\frac{1}{t}}}$

$=\underset{t\to 0}{lim}\sqrt{\frac{1-t\sin \frac{1}{t}}{1+t{\cos }^2\frac{1}{t}}}$

$\underset{t\to 0}{lim}t\sin \frac{1}{t}=0$ and $\underset{t\to 0}{lim}t{\cos }^2\frac{1}{t}=0$

$=\underset{t\to 0}{lim}\sqrt{\frac{1-0}{1+0}}$

$=1$

Hence, the vale of limit is $1$.