Evaluate the following limit: _ x 2 1+4x + 5+2x x-2 - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow2}\frac{\sqrt{1+4\text{x}}+\sqrt{5+2\text{x}}}{\text{x}-2}$

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Answer

$\lim\limits_{\text{x}\rightarrow2}\frac{\sqrt{1+4\text{x}}+\sqrt{5+2\text{x}}}{\text{x}-2}$
$=\lim\limits_{\text{x}\rightarrow2}\frac{\big(\sqrt{1+4\text{x}}-\sqrt{5+2\text{x}}\big)}{(\text{x}-2)}\times\frac{\big(\sqrt{1+4\text{x}}+\sqrt{5+2\text{x}}\big)}{\big(\sqrt{1+4\text{x}}+\sqrt{5+2\text{x}}\big)}$
$=\lim\limits_{\text{x}\rightarrow2}\frac{(1+4\text{x})-(5+2\text{x})}{(\text{x}-2)\big(\sqrt{1+4\text{x}}+\sqrt{5+2\text{x}}\big)}$
$=\lim\limits_{\text{x}\rightarrow2}\frac{-4+2\text{x}}{(\text{x}-2)\big(\sqrt{1+4\text{x}}+\sqrt{5+2\text{x}}\big)}$
$=\frac{2}{\sqrt{1+8}+\sqrt{5+4}}=\frac{2}{\sqrt{9}+\sqrt{9}}$
$=\frac{2}{6}=\frac13$

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