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Limits — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSLimits4 Marks
Question
Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow0}\frac{\sqrt{\text{a}+\text{x}}-\sqrt{\text{a}}}{\text{x}\sqrt{\text{a}^2+\text{ax}}}$
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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\sqrt{\text{a}+\text{x}}-\sqrt{\text{a}}}{\text{x}\sqrt{\text{a}^2+\text{ax}}}$$=\lim\limits_{\text{x}\rightarrow0}\frac{\big(\sqrt{\text{a}+\text{x}}-\sqrt{\text{a}}\big)}{\text{x}\sqrt{\text{a}^2+\text{ax}}}\times\frac{\big(\sqrt{\text{a}+\text{x}}+\sqrt{\text{a}}\big)}{\big(\sqrt{\text{a}+\text{x}}+\sqrt{\text{a}}\big)}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{(\text{a}+\text{x})-\text{a}}{\text{x}\sqrt{\text{a}^2+\text{ax}}\big(\sqrt{\text{a}+\text{x}}+\sqrt{\text{a}}\big)}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{\text{x}}{\text{x}\sqrt{\text{a}^2+\text{ax}}\big(\sqrt{\text{a}+\text{x}}+\sqrt{\text{a}}\big)}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{1}{\big(\sqrt{\text{a}^2+\text{ax}}\big)\big(\sqrt{\text{a}+\text{x}}+\sqrt{\text{a}}\big)}$
$=\frac{1}{\text{a}\big(\sqrt{2\text{a}}\big)}$
$=\frac{1}{2\text{a}\sqrt{\text{a}}}$

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