Question

$\text{The value of}\underset{x\to \infty }{lim}[\sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x}]\text{is}.$

• A. 1/2
• B. 1
• C. 0
• D. -1/2

Answer 1

The correct option is A

1/2

$=\underset{x\to \infty }{lim}[\sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x}]=\underset{x\to \infty }{lim}\frac{x+\sqrt{x+\sqrt{x}}-x}{\sqrt{x+\sqrt{x+\sqrt{x}}+\sqrt{x}}}=\underset{x\to \infty }{lim}\frac{\sqrt{x+\sqrt{x}}}{\sqrt{x+\sqrt{x+\sqrt{x}}+\sqrt{x}}}=\underset{x\to \infty }{lim}\frac{\sqrt{x}{(1+\frac{1}{\sqrt{x}})}^{1/2}}{\sqrt{x}[{(1+\frac{1}{\sqrt{x}}\sqrt{1+\frac{1}{\sqrt{x}}})}^{1/2}+1]}=\frac{1}{1+1}=\frac{1}{2}$