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

Question

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

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Answer

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

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