Question

Evaluate the limit $\underset{x\to 9}{lim}\frac{\log (x-9)}{\log (e^x-e^9)}$

Answer 1

$\underset{x\to 9}{lim}\frac{\log \left(x-9\right)}{\log \left(e^x-e^9\right)}$ ($\frac{0}{0}$form)

$=\underset{x\to 9}{lim}\frac{\frac{1}{x-9}}{\frac{e^x}{e^x-e^9}}$ ($\frac{0}{0}$ form)

$=\underset{x\to 9}{lim}\frac{e^x-e^9}{e^x\left(x-9\right)}$ ($\frac{0}{0}$form)

$=\underset{x\to 9}{lim}\frac{e^x}{e^x\left(x-9\right)+e^x}$

$=\underset{x\to 9}{lim}\frac{1}{x-9+1}$

$=1$