Question

${lim}_{x\to \infty }\frac{\sqrt{x^2+si{n}^{2}x}}{x+cosx}$ equals

• A. $1$
• B. $0$
• C. $\infty$
• D. does not exist

Answer 1

The correct option is A $1$

Given,

${lim}_{x\to \infty }\left(\frac{\sqrt{x^2+{\sin }^2\left(x\right)}}{x+\cos \left(x\right)}\right)$

dividing by highest denominator power

$={lim}_{x\to \infty }\left(\frac{\sqrt{1+\frac{{\sin }^2\left(x\right)}{x^2}}}{1+\frac{\cos \left(x\right)}{x}}\right)$

$=\frac{{lim}_{x\to \infty }\left(\sqrt{1+\frac{{\sin }^2\left(x\right)}{x^2}}\right)}{{lim}_{x\to \infty }(1+\frac{\cos \left(x\right)}{x})}$

$=\frac{\sqrt{1+0}}{1+0}$

$=1$