I want to show more concretely that
$$\lim_{x\to\infty} \frac{e^\sqrt{x}}{x^{a}} = \infty$$
To do this I did the natural log of the numerator and denominator and then did l'Hospital's rule.
My question is: it allowed to do the natural log of the numerator and denominator in order to put it in a clear form before evaluating the limit?
Cause I know $6/3=2$, but $\ln\left(6\right)/\ln\left(3\right)$ is not. If it is not allowed what is a way to concretely evaluate this limit.
Best Answer
$$\frac{e^{2x}}{e^x}$$ is a counterexample. After taking the natural logarithm, we get $2$, so the limit is $2$. The original limit is $\infty$.