Evaluate the following limit: _ x 1 3+x - 5-x x^2-1 - Vidyadip

Question

Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow1}\frac{\sqrt{3+\text{x}}-\sqrt{5-\text{x}}}{\text{x}^2-1}$

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Answer

$\lim\limits_{\text{x}\rightarrow1}\frac{\sqrt{3+\text{x}}-\sqrt{5-\text{x}}}{\text{x}^2-1}$$=\lim\limits_{\text{x}\rightarrow1}\frac{\big(\sqrt{3+\text{x}}-\sqrt{5-\text{x}}\big)}{\big(\text{x}^2-1\big)}\times\frac{\big(\sqrt{3+\text{x}}+\sqrt{5-\text{x}}\big)}{\big(\sqrt{3+\text{x}}+\sqrt{5-\text{x}}\big)}$
$=\lim\limits_{\text{x}\rightarrow1}\frac{(3+\text{x})-(5-\text{x})}{(\text{x}-1)(\text{x}+1)\big(\sqrt{3+\text{x}}+\sqrt{5+\text{x}}\big)}$
$=\lim\limits_{\text{x}\rightarrow1}\frac{-2+2\text{x}}{(\text{x}-1)(\text{x}+1)\big(\sqrt{3+\text{x}}+\sqrt{5+\text{x}}\big)}$
$=\lim\limits_{\text{x}\rightarrow1}\frac{2}{(\text{x}+1)\big(\sqrt{3+\text{x}}+\sqrt{5+\text{x}}\big)}$
$=\frac{2}{(1+1)\big(\sqrt{3+1}+\sqrt{5-1}\big)}=\frac{2}{(2)(2+2)}$
$=\frac14$

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