Can the limit of a quotient of trig functions approach a specific value?

[Jonas]

Homework Statement

Can lim x-> infinity sin(ln(x)) /cos(sqrt(x)) be evaluated with L'Hôpitals rule?

Relevant Equations

lim x→∞ sin(ln(x))/cos(√x)

Hello.
Sin and cos separately oscillates between [-1,1] so the limit of each as x approach infinity does not exist.
But can a quotient of the two acutally approach a certain value?
lim x→∞ sin(ln(x))/cos(√x) has to be rewritten if L'hôp. is to be applied but i can't seem to find a way to rewrite it to get a meaningful answer. Ofc i tried wolframalpha who states that it approaches + and - infinity which made me even more confused since i would normally interpret that as the limit does not exist?

 

[PeroK]

The function isn't defined where the denominator is zero. And, in any case, if you analyse those points you'll see the problem.

It's simpler if you let $y = \sqrt x$ and take $y \rightarrow \infty$.

 

[FactChecker]

You should look at the specific conditions where L'Hopital's rule can be used and see if those conditions are met. I suggest that you carefully review the statement of L'Hopital's rule. If you find one condition that is not met, state that condition and show that it is not met.