calculus - How to solve non-fractional limits like $lim_{x to infty} x^{(ln5) div (1+ln x)}$?

I have the limit $\lim_{x \to \infty} x^{(\ln5) \div (1+\ln x)}$. I am trying to figure out how to solve this, but I only know how to handle limits when they can be made into fractions. Is there some way I can do that here, or should I try something else?

This is a homework problem, and I don't need a complete answer, but I'd appreciate some advice.

Answer

We asked to evaluate $\lim \limits_{x \rightarrow \infty} x^{\frac{\ln(5)}{1+\ln(x)}}$ which can also be written as $\lim \limits_{x \rightarrow \infty} e^{\ln(x){\frac{\ln(5)}{1+\ln(x)}}}$. Using a few other rules, you can get to the following expression

$$e^{\ln(5) \cdot \lim \limits_{x \rightarrow \infty} \frac{\ln(x)}{1+\ln(x)} }$$

Now I will leave the details to you but as a hint, remind yourself of L'hospital's rule.

And finally, you can show:

$$\lim \limits_{x \rightarrow \infty} x^{\frac{\ln(5)}{1+\ln(x)}} = 5$$