Evaluate the following limit: _ x 3 x^2-3 x^2+33x-12 - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow\sqrt{3}}\frac{\text{x}^2-3}{{\text{x}^2+3\sqrt{3}\text{x}-12}}$

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Answer

$\lim\limits_{\text{x}\rightarrow\sqrt{3}}\frac{\text{x}^2-3}{{\text{x}^2+3\sqrt{3}\text{x}-12}}$
$=\lim\limits_{\text{x}\rightarrow\sqrt{3}}\frac{\big(\text{x}-\sqrt{3}\big)\big(\text{x}+\sqrt{3}\big)}{\text{x}^2+4\sqrt{3\text{x}}-\sqrt{3\text{x}}-12}$
$=\lim\limits_{\text{x}\rightarrow\sqrt{3}}\frac{\big(\text{x}-\sqrt{3}\big)\big(\text{x}+\sqrt{3}\big)}{\text{x}\big(\text{x}+4\sqrt{3}\big)-\sqrt{3}\big(\text{x}+4\sqrt{3}\big)}$
$=\lim\limits_{\text{x}\rightarrow\sqrt{3}}\frac{\big(\text{x}-\sqrt{3}\big)\big(\text{x}+\sqrt{3}\big)}{\big(\text{x}-\sqrt{3}\big)\big(\text{x}+4\sqrt{3}\big)}$
$=\frac{\sqrt{3}+\sqrt{3}}{\sqrt{3}+4\sqrt{3}}=\frac{2\sqrt{3}}{5\sqrt{3}}$
$=\frac25$

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