Consider $\lim_{x\to0}\frac{(1+x)^{1/x}-e}{x}$.
At first glance, it seems like the given limit is not in $\frac{0}{0}$ or $\frac{\infty}{\infty}$ form. (It is $\frac{1^{\infty}-e}{0}$ form.) But I know that $\lim_{x\to0}(1+x)^{1/x} = e$, and therefore, $\lim_{x\to0}\frac{(1+x)^{1/x}-e}{x}$ is in $\frac{e-e}{0} = \frac{0}{0}$ form. In such a scenario, can I apply L'Hospital's rule to evaluate the limit?
I applied L'Hospital's rule and managed to evaluate the limit correctly, but I'm not sure whether it is valid to apply L'Hospital's rule here.