Evaluate the following limit: _ x 1 (2x-3) ( x -1 ) 3x^2+3x-6 - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow1}\frac{(2\text{x}-3)\big(\sqrt{\text{x}}-1\big)}{3\text{x}^2+3\text{x}-6}$

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Answer

$\lim\limits_{\text{x}\rightarrow1}\frac{(2\text{x}-3)\big(\sqrt{\text{x}}-1\big)}{3\text{x}^2+3\text{x}-6}$
$=\lim\limits_{\text{x}\rightarrow1}\frac{(2\text{x}-3)\big(\sqrt{\text{x}}-1\big)}{3\big[\text{x}^2+\text{x}-2\big]}$
$=\lim\limits_{\text{x}\rightarrow1}\frac{(2\text{x}-3)\big(\sqrt{\text{x}}-1\big)}{3\big[\text{x}^2+2\text{x}-\text{x}-2\big]}$
$=\lim\limits_{\text{x}\rightarrow1}\frac{(2\text{x}-3)\big(\sqrt{\text{x}}-1\big)}{3(\text{x}+2)(\text{x}-1)}$
$=\lim\limits_{\text{x}\rightarrow1}\frac{(2\text{x}-3)\big(\sqrt{\text{x}}-1\big)}{3(\text{x}+2)\big(\sqrt{\text{x}-1}\big)\big(\sqrt{\text{x}+1}\big)}$
$=\lim\limits_{\text{x}\rightarrow1}\frac{(2\text{x}-3)}{3(\text{x}+2)\big(\sqrt{\text{x}}+1\big)}$
$=\frac{(2-3)}{3(1+2)(1+1)}$
$=\frac{-1}{3\times3\times2}$
$=-\frac{1}{18}$

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