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  4. The value of displaystyle lim x arrow ∞(√x+√x+√x-√x) is

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Q. The value of $\displaystyle\lim _{x \rightarrow \infty}(\sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x})$ is

Limits and Derivatives

Solution:

$\displaystyle\lim _{x \rightarrow \infty}[\sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x}]$
$=\displaystyle\lim _{x \rightarrow \infty} \frac{x+\sqrt{x+\sqrt{x}}-x}{\sqrt{x+\sqrt{x+\sqrt{x}}+\sqrt{x}}}$
$=\displaystyle\lim _{x \rightarrow \infty} \frac{\sqrt{x+\sqrt{x}}}{\sqrt{x+\sqrt{x+\sqrt{x}}+\sqrt{x}}}$
$=\displaystyle\lim _{x \rightarrow \infty} \frac{\sqrt{x}}{\sqrt{x}\left[\left(1+\frac{1}{\sqrt{x}} \sqrt{1+\frac{1}{\sqrt{x}}}\right)^{1 / 2}+1\right]^{1 / 2}}$
$=\frac{1}{1+1}=\frac{1}{2}$