Evaluate: $\underset{x→e}{\lim}\frac{\log x-1}{x-e}$.
Answer & explanation
Correct answer: option 3
$\underset{x→e}{\lim}\frac{ln\, x-1}{x-e}=\underset{x→e}{\lim}\frac{ln\,x-ln\,e}{x-e}=\underset{x→e}{\lim}\frac{ln(x/e)}{x-e}=\underset{x→e}{\lim}\frac{ln(1+\frac{x-e}{e})}{\frac{x-e}{e}.e}$
$=\underset{x→e}{\lim}\frac{ln(1+\frac{x-e}{e})^{e/x-e}}{e}=\frac{1}{e}$