Question

If the function $f\left(x\right)=\frac{\log x-1}{x-e}$, for $x\ne e$ is continuous at $x=e$, then find $f\left(e\right)$.

Answer 1

As f is continuous, $\underset{x\to e}{lim}f\left(x\right)=f\left(e\right)$

$\underset{x\to e}{lim}f\left(x\right)=\underset{x\to e}{lim}\frac{lo{g}_x-1}{x-e}$ is $\frac{0}{0}$ form, applying l' opital

$\underset{x\to e}{lim}\frac{\frac{1}{x}-0}{1-e}=\frac{1}{e}=f\left(e\right)$.