The value of $\underset{x\to e}{lim} \frac{\log x-1}{x-e},$ is

Solution:

We have,

$\underset{x\to e}{lim} \frac{\log x-1}{x-e}=\underset{x\to e}{lim} \frac{\log x-\log e}{x-e}=\underset{x\to e}{lim} \frac{\log \frac{x}{e}}{x-e}$

$=\underset{x\to e}{lim} \frac{\log \left(1+\frac{e^x-1}{e}\right)}{x-e}=\underset{x\to e}{lim} \frac{\log \left\{1+\left(\frac{x-e}{e}\right)\right\}}{e\left(\frac{x-e}{e}\right)}=\frac{1}{e}$