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

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

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