Evaluate the following limit: _ x 2 (1 x-2 -4 x^3-2x^2 ) - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow2}\Big(\frac{1}{\text{x}-2}-\frac{4}{\text{x}^3-2\text{x}^2}\Big)$

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Answer

$\lim\limits_{\text{x}\rightarrow2}\Big(\frac{1}{\text{x}-2}-\frac{4}{\text{x}^3-2\text{x}^2}\Big)$
$=\lim\limits_{\text{x}\rightarrow2}\Big(\frac{1}{\text{x}-2}-\frac{4}{\text{x}^2(\text{x}-2)}\Big)$
$=\lim\limits_{\text{x}\rightarrow2}\Big(\frac{\text{x}^2-4}{\text{x}^2(\text{x}-2)}\Big)$
$=\lim\limits_{\text{x}\rightarrow2}\frac{(\text{x}-2)(\text{x}+2)}{\text{x}^2(\text{x}-2)}$
$=\lim\limits_{\text{x}\rightarrow2}\frac{(\text{x}+2)}{\text{x}^2}$
$=\frac{(2+2)}{2^2}=\frac{4}{4}$
$=1$

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