Question

$\underset{x\to \infty }{lim}\left(\frac{\sqrt{(1-\cos x)+\sqrt{(1-\cos x)+\sqrt{(}1-\cos x)+...\infty )-1}}}{x^2}\right)$ equals to

• A. 0
• B. $\frac{1}{2}$
• C. 1
• D. 2

Answer 1

Answer: (B)

$\frac{1}{2}$

Let $\sqrt{(1-\cos x)+\sqrt{(1-\cos x)+\sqrt{(1-\cos x)+...\infty )}}}=y$

$\Rightarrow y=\sqrt{(1-\cos x)+y}$

$\Rightarrow y^2-y+(\cos x-1)=0$

$\Rightarrow y=\frac{1+\sqrt{5-4\cos x}}{2}$

Now required limit is ${lim}_{x\to 0}$ $\frac{1+\sqrt{5-4cosx-1}}{2x^2}$ $=\frac{1}{2}$