Evaluate the following limit: _ x a logx-loga x-a - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{logx}-\text{loga}}{\text{x}-\text{a}}$

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Answer

$\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{logx}-\text{loga}}{\text{x}-\text{a}}$
$=\lim\limits_{\text{x}\rightarrow\text{a}}\frac{\text{log}\frac{\text{x}}{\text{a}}}{\text{a}\Big(\frac{\text{x}}{\text{a}}-1\Big)}$
Let $\text{ h}=\frac{\text{x}}{\text{a}}-1$
$=\frac{1}{\text{a}}\lim\limits_{\text{x}\rightarrow\text{a}}\frac{\text{log(h+1)}}{\text{h}}$
$=\frac{1}{\text{a}}$

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