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

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow1}\Big(\frac{1}{\text{x}-1}-\frac{2}{\text{x}^2-1}\Big)$

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Answer

$\lim\limits_{\text{x}\rightarrow1}\Big(\frac{1}{\text{x}-1}-\frac{2}{\text{x}^2-1}\Big)$
$=\lim\limits_{\text{x}\rightarrow1}\Big(\frac{1}{\text{x}-1}-\frac{2}{(\text{x}-1)(\text{x}+1)}\Big)$
$=\lim\limits_{\text{x}\rightarrow1}\Big(\frac{\text{x}+1-2}{(\text{x}-1)(\text{x}+1)}\Big)$
$=\lim\limits_{\text{x}\rightarrow1}\Big(\frac{\text{x}-1}{(\text{x}-1)(\text{x}+1)}\Big)$
$=\lim\limits_{\text{x}\rightarrow1}\Big(\frac{1}{(\text{x}+1)}\Big)$
$=\frac{1}{1+1}=\frac12$

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