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

Question

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

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Answer

$\lim\limits_{\text{x}\rightarrow\infty}\sqrt{\text{x}}\Big\{\sqrt{\text{x}+1}-\sqrt{\text{x}}\Big\}$$=\lim\limits_{\text{x}\rightarrow\infty}\Big(\sqrt{\text{x}^2+\text{x}}-{\text{x}}\Big)$
$=\lim\limits_{\text{n}\rightarrow\infty}\Bigg(\big(\sqrt{\text{x}^2+\text{x}}-\text{x}\big)\times\frac{\big(\sqrt{\text{x}^2+\text{x}}+\text{x}\big)}{\sqrt{\text{x}^2+\text{x}}+\text{x}}\Bigg)$
$=\lim\limits_{\text{n}\rightarrow\infty}\Bigg(\frac{\big(\text{x}^2+\text{x}\big)-\text{x}^2}{\sqrt{\text{x}^2+\text{x}}+\text{x}}\Bigg)$
$=\lim\limits_{\text{n}\rightarrow\infty}\bigg(\frac{\text{x}}{\sqrt{\text{x}^2+\text{x}}+\text{x}}\bigg)$ $\Big[\frac{\infty}{\infty}\text{ from}\Big]$
$=\lim\limits_{\text{n}\rightarrow{\infty}}\begin{pmatrix}\frac{1}{\sqrt{\frac{\text{x}^2}{\text{x}^2}}+\frac{\text{x}}{\text{x}^2}+1}\end{pmatrix}$
$=\lim\limits_{\text{n}\rightarrow{\infty}}\begin{pmatrix}\frac{1}{\sqrt{1+\frac{1}{\text{x}}}+1}\end{pmatrix}$
$=\frac{1}{1+1}$
$=\frac12$

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