Evaluate the following one sided limits: _ x 2^- x-3 x^2-4 . - Vidyadip

Question

Evaluate the following one sided limits:
$\lim\limits_{\text{x}\rightarrow2^-}\frac{\text{x}-3}{\text{x}^2-4}.$

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Answer

$\lim\limits_{\text{x}\rightarrow2^-}\frac{\text{x}-3}{(\text{x}^2-4)}$
$=\lim\limits_{\text{h}\rightarrow0}\frac{(2-\text{h})-3}{(2-\text{h)}^2-4^2}$ $\Big[\because\lim\limits_{\text{x}\rightarrow2^-}\text{f(x)}=\lim\limits_{\text{h}\rightarrow0}\text{f}(2-\text{h)}\Big]$
$=\lim\limits_{\text{h}\rightarrow0}\frac{(2-\text{h}-3)}{(2-\text{h}+2)(2-\text{h}-2)}$
$=\lim\limits_{\text{h}\rightarrow0}\frac{-1-\text{h}}{(4-\text{h})(-\text{h})}$
$=\lim\limits_{\text{h}\rightarrow0}\frac{\frac{1}{\text{h}}+1}{(4-\text{h})}$
$=\frac{\frac{1}{0}+1}{4}=\infty$

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