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

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow3}\Big(\frac{1}{\text{x}-3}-\frac{2}{\text{x}^2-4\text{x}+3}\Big)$

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Answer

$\lim\limits_{\text{x}\rightarrow3}\Big(\frac{1}{\text{x}-3}-\frac{2}{\text{x}^2-4\text{x}+3}\Big)$
$=\lim\limits_{\text{x}\rightarrow3}\Big(\frac{1}{\text{x}-3}-\frac{2}{(\text{x}-3)(\text{x}-1)}\Big)$
$=\lim\limits_{\text{x}\rightarrow3}\Big(\frac{\text{x}-1-2}{(\text{x}-1)(\text{x}-3)}\Big)$
$=\lim\limits_{\text{x}\rightarrow3}\bigg(\frac{\text{x}-3}{(\text{x}-1)(\text{x}-3)}\bigg)$
$=\lim\limits_{\text{x}\rightarrow3}\frac{1}{\text{x}-1}$
$=\frac{1}{3-1}$
$=\frac12$

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