Question

$\underset{x\to \infty }{lim}\frac{{\cos }^{2}x}{x}=$

• A. $0$
• B. $1$
• C. $-1$
• D. $\infty$

Answer 1

The correct option is A

$0$

We have, $\underset{x\to \infty }{lim}\frac{{\cos }^{2}x}{x}$
cosine graph is

How the graph of ${\cos }^{2}x$ looks? (Negative values of y will be eliminated)

$0\le {\cos }^{2}x\le 1$

Dividing throughout inequality by $x$.

$\frac{0}{x}\le \frac{co{s}^{2}x}{x}\le \frac{1}{x}$ $(x\ne 0)$

$0\le \frac{{\cos }^{2}x}{x}\le \frac{1}{x}$

Taking $\underset{x\to \infty }{lim}$ throughout the inequality
$\underset{x\to \infty }{lim}0\le \frac{{\cos }^{2}x}{x}\le \underset{x\to \infty }{lim}\frac{1}{x}$

Since, $\underset{x\to \infty }{lim}\left(0\right)=0$

& $\underset{x\to \infty }{lim}\frac{1}{x}=0$

Therefore $\underset{x\to \infty }{lim}\frac{{\cos }^{2}x}{x}=0$
(Using sandwich theorm)