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

Question

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

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Answer

$\lim\limits_{\text{x}\rightarrow2}\Big(\frac{\text{x}}{\text{x}-2}-\frac{4}{\text{x}^2-2\text{x}}\Big)$
$=\lim\limits_{\text{x}\rightarrow2}\Big(\frac{\text{x}}{\text{x}-2}-\frac{4}{\text{x}(\text{x}-2)}\Big)$
$=\lim\limits_{\text{x}\rightarrow2}\Big(\frac{\text{x}(\text{x}-4)}{\text{x}(\text{x}-2)}\Big)$
$=\lim\limits_{\text{x}\rightarrow2}\Big(\frac{\text{x}^2-4}{\text{x}(\text{x}-2)}\Big)$
$=\lim\limits_{\text{x}\rightarrow2}\frac{(\text{x}-2)(\text{x}+2)}{\text{x}(\text{x}-2)}$
$=\lim\limits_{\text{x}\rightarrow2}\frac{(\text{x}+2)}{\text{x}}$
$=\frac{2+2}{2}$
$=\frac42$
$=2$

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