Evaluate the following limit: _ x 1 5x-4 - x x^3-1 - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow1}\frac{\sqrt{5\text{x}-4}-\sqrt{\text{x}}}{\text{x}^3-1}$

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Answer

$\lim\limits_{\text{x}\rightarrow1}\frac{\sqrt{5\text{x}-4}-\sqrt{\text{x}}}{\text{x}^3-1}$$=\lim\limits_{\text{x}\rightarrow1}\frac{\big(\sqrt{5\text{x}-4}-\sqrt{\text{x}}\big)}{(\text{x}-1)\big(\text{x}^2+1+1\big)}\times\frac{\big(\sqrt{5\text{x}-4}+\sqrt{\text{x}}\big)}{\big(\sqrt{5\text{x}-4}+\sqrt{\text{x}}\big)}$
$=\lim\limits_{\text{x}\rightarrow1}\frac{(5\text{x}-4-\text{x})}{(\text{x}-1)\big(\text{x}^2+1+\text{x}\big)\big(\sqrt{5\text{x}-4}+\sqrt{\text{x}}\big)}$
$=\lim\limits_{\text{x}\rightarrow1}\frac{4(\text{x}-1)}{(\text{x}-1)\big(\text{x}^2+\text{x}+1\big)\big(\sqrt{5\text{x}-4}+\sqrt{\text{x}}\big)}$
$=\frac{4}{(1+1+1)\big(\sqrt{5-4}+\sqrt{1}\big)}$
$=\frac{4}{(3)(1+1)}$
$=\frac{4}{3\times2}=\frac23$

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