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

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow2}\frac{\text{x}-2}{\text{log}_\text{a}(\text{x}-1)}$

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Answer

$\lim\limits_{\text{x}\rightarrow2}\frac{\text{x}-2}{\text{log}_\text{a}(\text{x}-1)}$
Let x - 2 = h
$=\lim\limits_{\text{h}\rightarrow0}\frac{\text{h}}{\text{log}_\text{a}(\text{h+1})}$
$=\lim\limits_{\text{h}\rightarrow0}\frac{\frac{\text{log a}}{\text{log(h+1)}}}{\text{h}}$
$=\text{log a}$

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