Chapter 3.5, Problem 57E

Finding Natural Logarithms as Limits.

Given that the derivative of $f\left(x\right)=a^x$ is $f\text{'}\left(x\right)=a^x\left(\ln a\right)$, in Section 3.1, we showed that

$f\text{'}\left(x\right)=a^x\cdot \underset{h\to 0}{\lim }\frac{a^h-1}{h}$.

Thus, we can define

$\ln a=\underset{h\to 0}{\lim }\frac{a^h-1}{h}$.

In Exercises 54-57, use this definition to find each limit.

$\underset{h\to 0}{\lim }\frac{\sqrt{e^h}-1}{h}$