_ x ( x+ x+x -x) is equal to - Vidyadip

MCQ

$\lim _{x \rightarrow \infty}(\sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x})$ is equal to

A
$0$
✓
$\frac{1}{2}$
C
$\log 2$
D
$e^4$

✓

Answer

Correct option: B.

$\frac{1}{2}$

(B)
$\lim _{x \rightarrow \infty}(\sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x})$
$=\lim _{x \rightarrow \infty} \frac{x+\sqrt{x+\sqrt{x}}-x}{\sqrt{x+\sqrt{x+\sqrt{x}}}+\sqrt{x}}$
$=\lim _{x \rightarrow \infty} \frac{\sqrt{x+\sqrt{x}}}{\sqrt{x+\sqrt{x+\sqrt{x}}}+\sqrt{x}}$
$=\lim _{x \rightarrow \infty} \frac{\sqrt{1+x^{-1 / 2}}}{\sqrt{1+\sqrt{x^{-1}+x^{-3 / 2}}}+1}=\frac{1}{2}$

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