Question

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

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

Answer 1

The correct option is B $1$

$\underset{x\to \infty }{lim}\sqrt{\frac{x+sinx}{x-cosx}}$
$=\underset{x\to \infty }{lim}\frac{x^{\frac{1}{2}}\sqrt{1+\frac{sinx}{x}}}{x^{\frac{1}{2}}\sqrt{1-\frac{cosx}{x}}}$
We know, that for any value of x, sinx and cosx will be [-1,1]
So, $=\underset{x\to \infty }{lim}\frac{Sinx}{x}=0$
And $=\underset{x\to \infty }{lim}\frac{Cosx}{x}=0$
$=x^{\frac{1}{2}}\sqrt{1+\frac{sinx}{x}}\frac{\sqrt{1+\frac{sinx}{x}}}{\sqrt{1+\frac{cosx}{x}}}$
$=\underset{x\to \infty }{lim}\frac{1}{1}$
$=1$