I'd like more of an explanation than a solution, I'm sorry but I'm studying math again after 12 years, and I don't understand basic concepts.
I have to evaluate some limits without using L'Hopital rule, just by definition. So I got:
$$\lim\limits_{h\to \:0}\left(\frac{e^h-1}{h}\right).$$
After evaluating it in a Math app, it states that it's just 1 due to the common limit evaluation, but I don't understand why it's 1.
Depends what your definition of $e^h$ is.
If you know the power series ($e^h =\sum_{n=0}^{\infty} \frac{h^n}{n!}$), it is easy.
If you know that $(e^h)' = e^h$, then $e^h-1 =\int_0^h e^x dx $, and you can use the mean value theorem combined with $e^0=1$ and the continuity of $e^h$ to get the result.
There are probably a few other ways, but these occurred to me.