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

Question

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

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Answer

$\lim\limits_{\text{x}\rightarrow1}\frac{\text{x}^2-\sqrt{\text{x}}}{\sqrt{\text{x}}-1}$$=\lim\limits_{\text{x}\rightarrow1}\frac{\big(\text{x}^2-\sqrt{\text{x}}\big)\big(\text{x}^2+\sqrt{\text{x}}\big)}{\big(\sqrt{\text{x}}-1\big)\big(\text{x}^2+\sqrt{\text{x}}\big)}$
$=\lim\limits_{\text{x}\rightarrow1}\frac{\text{x}^4-\text{x}}{\big(\sqrt{\text{x}-1}\big)\big({\text{x}^2}+\sqrt{\text{x}}\big)}$
$=\lim\limits_{\text{x}\rightarrow1}\frac{\text{x}\big(\text{x}^3-1\big)}{\big(\sqrt{\text{x}-1}\big)\big(\text{x}^2+\sqrt{\text{x}}\big)}$
$=\lim\limits_{\text{x}\rightarrow1}\frac{\text{x}(\text{x}-1)\big(\text{x}^2+1+\text{x}\big)}{\big(\sqrt{\text{x}-1}\big)\big(\text{x}^2+\sqrt{\text{x}}\big)}$
$=\lim\limits_{\text{x}\rightarrow1}\frac{\text{x}\big(\sqrt{\text{x}+1}\big)\big({\text{x}^2+1+\text{x}}\big)}{\big(\text{x}^2+\sqrt{\text{x}}\big)}$
$=\frac{1(1+1)(1+1+1)}{1+1}$
$=\frac62$
$=3$

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