Question

The value of $\underset{x\to \infty }{lim}\frac{5^{\sin x}+2x+1}{\cos x-\sqrt{1+x^2}}$ is

• A. $-2$
• B. $1$
• C. $-1$
• D. $2$

Answer 1

The correct option is A $-2$

$\underset{x\to \infty }{lim}\frac{5^{\sin x}+2x+1}{\cos x-\sqrt{1+x^2}}=\underset{x\to \infty }{lim}\frac{\frac{(5^{\sin x}+2x+1)}{x}}{\frac{(\cos x-\sqrt{1+x^2})}{x}}$
$=\underset{x\to \infty }{lim}\frac{\frac{5^{\sin x}}{x}+2+\frac{1}{x}}{\frac{\cos x}{x}-\sqrt{\frac{1}{x^2}+1}}=\frac{2}{-\sqrt{1}}=\frac{2}{-1}=-2$