Evaluate the following limits: _ x x^2+7x -x - Vidyadip

Question

Evaluate the following limits:
$\lim\limits_{\text{x}\rightarrow\infty}\sqrt{\text{x}^2+7\text{x}}-\text{x}$

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Answer

$\lim\limits_{\text{x}\rightarrow\infty}\sqrt{\text{x}^2+7\text{x}}-\text{x}$
$=\lim\limits_{\text{x}\rightarrow\infty}\Bigg(\frac{\big(\sqrt{\text{x}^2+7\text{x}}-\text{x}\big)\big(\sqrt{\text{x}^2+7\text{x}}+\text{x}\big)}{\sqrt{\text{x}^2+7\text{x}}+\text{x}}\Bigg)$
$=\lim\limits_{\text{x}\rightarrow\infty}\Bigg(\frac{\big({\text{x}^2+7\text{x}}\big)-{\text{x}}}{\sqrt{\text{x}^2+7\text{x}}+\text{x}}\Bigg)$
$=\lim\limits_{\text{x}\rightarrow\infty}\frac{{7\text{x}}}{\sqrt{\text{x}^2+7\text{x}}+\text{x}}$
$=\lim\limits_{\text{x}\rightarrow\infty}\frac{{7}}{\sqrt{\frac{\text{x}^2}{\text{x}^2}\frac{7\text{x}}{\text{x}^2}}+1}$
$=\lim\limits_{\text{x}\rightarrow\infty}\frac{{7}}{\sqrt{1+\frac{7}{\text{x}}}+1}$
$=\frac72$

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