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Limits — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsLimits5 Marks
Question
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow1}\frac{\sqrt{5\text{x}-4}-\sqrt{\text{x}}}{\text{x}-1}$
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Answer

$\lim\limits_{\text{x}\rightarrow1}\frac{\sqrt{5\text{x}-4}-\sqrt{\text{x}}}{\text{x}-1}$
$=\lim\limits_{\text{x}\rightarrow1}\frac{\big(\sqrt{5\text{x}-4}-\sqrt{\text{x}}\big)}{\text{x}-1}\times\frac{\big(\sqrt{5\text{x}-4}+\sqrt{\text{x}}\big)}{\big(\sqrt{5\text{x}-4}+\sqrt{\text{x}}\big)}{}$
$=\lim\limits_{\text{x}\rightarrow1}\frac{((5\text{x}-4)-\text{x)}}{(\text{x}-1)\big(\sqrt{5\text{x}-4}+\sqrt{\text{x}}\big)}$
$=4\lim\limits_{\text{x}\rightarrow1}\frac{(\text{x}-1)}{(\text{x}-1)\big(\sqrt{5\text{x}-4}+\sqrt{\text{x}}\big)}$
$=4\lim\limits_{\text{x}\rightarrow1}\frac{1}{\big(\sqrt{5\text{x}-4}+\sqrt{\text{x}}\big)}$
$=4\times\frac{1}{\sqrt{5-4}+\sqrt{1}}$
$=4\times\frac{1}{\sqrt{5-4}+\sqrt{1}}$
$=4\times\frac{1}{\sqrt{1}+\sqrt{1}}$
$=\frac{4}{2}=2$

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