Evaluate the following limit: _ x 5 x^2-9x+20 x^2 -6x+5 - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow5}\frac{\text{x}^2-9\text{x}+20}{{\text{x}^2}-6\text{x}+5}$

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Answer

$\lim\limits_{\text{x}\rightarrow5}\frac{\text{x}^2-9\text{x}+20}{{\text{x}^2}-6\text{x}+5}$
$=\lim\limits_{\text{x}\rightarrow5}\frac{\text{x}^2-4\text{x}-5\text{x}+20}{\text{x}^2+\text{x}-5\text{x}+5}$
$=\lim\limits_{\text{x}\rightarrow5}\frac{\text{x}(\text{x}-4)-5(\text{x}-4)}{\text{x}(\text{x}-1)-5(\text{x}-1)}$
$=\lim\limits_{\text{x}\rightarrow5}\frac{(\text{x}-5)(\text{x}-4)}{(\text{x}-5)(\text{x}-1)}$
$=\lim\limits_{\text{x}\rightarrow5}\frac{\text{x}-4}{\text{x}-1}$
$=\frac{5-4}{5-1}$
$=\frac14$

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