Evaluate the following one sided limits: _ x 0^- x^2-3x+2 x^3-2x^…

Question

Evaluate the following one sided limits:
$\lim\limits_{\text{x}\rightarrow0^-}\frac{\text{x}^2-3\text{x}+2}{\text{x}^3-2\text{x}^2}$

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Answer

$\lim\limits_{\text{x}\rightarrow0^-}\frac{\text{x}^2-3\text{x}+2}{\text{x}^3-2\text{x}^2}$$=\lim\limits_{\text{x}\rightarrow0^-}\frac{\text{x}^2-\text{x}-2\text{x}+2}{\text{x}^2(\text{x}-2)}$
$=\lim\limits_{\text{x}\rightarrow0^-}\frac{\text{x}(\text{x}-1)-2(\text{x}-1)}{\text{x}^2(\text{x}-2)}$
$=\lim\limits_{\text{x}\rightarrow0^-}\frac{(\text{x}-1)(\text{x}-2)}{\text{x}^2(\text{x}-2)}$
$=\lim\limits_{\text{x}\rightarrow0^-}\frac{(\text{x}-1)}{\text{x}^2}$
$=\lim\limits_{\text{h}\rightarrow0}\frac{(0-\text{h}-1)}{(0-\text{h})^2}$
$\Rightarrow\frac{-\text{h}}{\text{h}^2}=\frac{-1}{\text{h}}=\frac{-1}{0}=-\infty$

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