Evaluate the following limit: _ x x+1 - x - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow{\infty}}{\sqrt{\text{x}+1}-\sqrt{\text{x}}}{}$

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Answer

$\lim\limits_{\text{x}\rightarrow{\infty}}{\sqrt{\text{x}+1}-\sqrt{\text{x}}}{}$
$=\lim\limits_{\text{x}\rightarrow{\infty}}\frac{\big(\sqrt{\text{x}+1}-\sqrt{\text{x}}\big)\big(\sqrt{\text{x}+1}+\sqrt{\text{x}}\big)}{\big(\sqrt{\text{x}+1}+\sqrt{\text{x}}\big)}$
$=\lim\limits_{\text{x}\rightarrow{\infty}}\frac{\big(\text{x}+1-\text{x}\big)}{\sqrt{\text{x}+1}+\sqrt{\text{x}}}$
$=\lim\limits_{\text{x}\rightarrow{\infty}}\bigg(\frac{1}{\sqrt{\text{x}+1}+\sqrt{\text{x}}}\bigg)$
$=\frac{1}{\infty}$
$=0$

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