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Limits — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSLimits2 Marks
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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