Evaluate the following limit: _ x 5 x-5 6x-5 - 4x+5 - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow5}\frac{\text{x}-5}{\sqrt{6\text{x}-5}-\sqrt{4\text{x}+5}}$

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Answer

$\lim\limits_{\text{x}\rightarrow5}\frac{\text{x}-5}{\sqrt{6\text{x}-5}-\sqrt{4\text{x}+5}}$
$=\lim\limits_{\text{x}\rightarrow5}\frac{\text{x}-5}{\big(\sqrt{6\text{x}-5}-\sqrt{4\text{x}+5}\big)}\times\frac{\big(\sqrt{6\text{x}-5}+\sqrt{4\text{x}+5}\big)}{\big(\sqrt{6\text{x}-5}+\sqrt{4\text{x}+5}\big)}$
$=\lim\limits_{\text{x}\rightarrow5}\frac{(\text{x}-5)\big(\sqrt{6\text{x}-5}+\sqrt{4\text{x}+5}\big)}{(6\text{x}-5)-(4\text{x}+5)}$
$=\lim\limits_{\text{x}\rightarrow5}\frac{(\text{x}-5)\big(\sqrt{6\text{x}-5}+\sqrt{4\text{x}+5}\big)}{2\text{x}-10}$
$=\lim\limits_{\text{x}\rightarrow5}\frac{(\text{x}-5)\big(\sqrt{6\text{x}-5}+\sqrt{4\text{x}+5}\big)}{2(\text{x}-5)}$
$=\frac{\sqrt{6(5)-5}+\sqrt{4(5)+5}}{2}$
$=\frac{\sqrt{25}+\sqrt{25}}{2}$
$=\frac{5+5}{2}=5$

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