Evaluate the following limit: _ x -2 x^3+x^2+4x+12 x^3+3x+2 - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow-2}\frac{\text{x}^3+\text{x}^2+4\text{x}+12}{\text{x}^3+3\text{x}+2}$

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Answer

$\lim\limits_{\text{x}\rightarrow-2}\frac{\text{x}^3+\text{x}^2+4\text{x}+12}{\text{x}^3+3\text{x}+2}$
$=\lim\limits_{\text{x}\rightarrow-2}\frac{(\text{x}+2)\big(\text{x}^2+\text{x}+6\big)}{(\text{x}+2)\big(\text{x}^2-2\text{x}+1\big)}$
$=\lim\limits_{\text{x}\rightarrow-2}\frac{\text{x}^2+\text{x}+6}{\text{x}^2-2\text{x}+1}$
$=\frac{(-2)^2-(-2)+6}{(-2)^2-2(-2)+1}$
$=\frac{4+2+6}{4+4+1}$
$=\frac{12}{9}$
$=\frac{4}{3}$

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