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

Question

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

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Answer

$\lim\limits_{\text{x}\rightarrow2}\Big(\frac{1}{\text{x}-2}-\frac{4}{\text{x}^3-2\text{x}}\Big)$
$=\lim\limits_{\text{x}\rightarrow2}\Big(\frac{1}{\text{x}-2}-\frac{2}{\text{x}(\text{x}-2)}\Big)$
$=\lim\limits_{\text{x}\rightarrow2}\Big(\frac{(\text{x}-2)}{(\text{x}-2)(\text{x})}\Big)$
$=\lim\limits_{\text{x}\rightarrow2}\Big(\frac{(\text{x}-2)}{(\text{x}-2)(\text{x})}\Big)$
$=\lim\limits_{\text{x}\rightarrow2}\frac{1}{\text{x}}$
$=\frac12$

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