Question

$\underset{x\to a}{\lim }\frac{\log x-\log a}{x-a}$

Answer 1

$$\begin{gathered} \underset{x\to a}{\lim }\left[\frac{\log x-\log a}{x-a}\right] \\ =\underset{x\to a}{\lim }\left[\frac{\log \left({\displaystyle \frac{x}{a}}\right)}{a\left({\displaystyle \frac{x}{a}}-1\right)}\right] \\ =\underset{x\to a}{\lim }\left[\frac{\log \left[1+\left({\displaystyle \frac{x}{a}}-1\right)\right]}{a\left({\displaystyle \frac{x}{a}}-1\right)}\right] \\ x\to a \\ \therefore \frac{x}{a}\to 1 \\ \Rightarrow \frac{x}{a}-1\to 0 \\ \mathrm{Let}y=\frac{x}{a}-1 \\ x\to a \\ \therefore y\to 0 \\ =\underset{y\to 0}{\lim }\left[\frac{\log \left(1+y\right)}{a\times y}\right] \\ =\frac{1}{a}\times 1 \\ =\frac{1}{a} \end{gathered}$$