Evaluate the following limit: _ x x+1 - x x+2 - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow\infty}\Big\{\sqrt{\text{x}+1}-\sqrt{\text{x}}\Big\}\sqrt{\text{x}+2}$

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Answer

$\lim\limits_{\text{x}\rightarrow\infty}\Big\{\sqrt{\text{x}+1}-\sqrt{\text{x}}\Big\}\sqrt{\text{x}+2}$
$=\lim\limits_{\text{x}\rightarrow\infty}\text{x}\Big[\sqrt{\text{x}+1}-\sqrt{\text{x}}\Big]\times\frac{\big[\sqrt{\text{x}+1}+\sqrt{\text{x}}\big]}{\big[\sqrt{\text{x}+1}+\sqrt{\text{x}}\big]}\times\frac{\sqrt{\text{x}+2}\times\sqrt{\text{x}+2}}{\sqrt{\text{x}+2}}$
$=\lim\limits_{\text{x}\rightarrow\infty}\frac{(\text{x}+1-\text{x})}{\sqrt{\text{x}+1}+\sqrt{\text{x}}}\times\frac{(\text{x}+2)}{\sqrt{\text{x}+2}}$
$=\lim\limits_{\text{x}\rightarrow\infty}\frac{1(\text{x}+2)}{\text{x}\big(\sqrt{\text{x}+1}+\sqrt{{\text{x}}}\big)\big(\sqrt{\text{x}+2}\big)}$
$=\lim\limits_{\text{x}\rightarrow{\infty}}\frac{\text{x}\Big(1+\frac{2}{\text{x}}\Big)}{\sqrt{\text{x}}\bigg(\sqrt{1+\frac{1}{\text{x}}}+1\bigg)\bigg(\sqrt{1+\frac{2}{\text{x}}}\bigg)\sqrt{\text{x}}}$
$=\lim\limits_{\text{x}\rightarrow{\infty}}\frac{\Big(1+\frac{2}{\text{x}}\Big)}{\bigg(\sqrt{1+\frac{1}{\text{x}}}+\sqrt{1}\bigg)\bigg(\sqrt{1+\frac{2}{\text{x}}}\bigg){}}$
$=\lim\limits_{\text{x}\rightarrow\infty}\frac{(1+0)}{(1+1)\times1}=\frac12$

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