How to evaluate a limit

[kanta_opu]

Hello, konichiwa! I'm trying to evaluate this limit.
lim x→∞ {cos(1/x)}^x²
According to Wolfram Alpha, the answer is 1/sqrt(e), and now I want to know the process of evaluating this limit.
https://wolframalpha.com/input/?i=lim+x→∞+{cos(1/x)}^x^2
The expression above might not be readable, so please use the link above to see clear one.

 

[MarkFL]

Hello, and welcome to FMH!

I would begin by writing:

[MATH]L=\lim_{x\to\infty}\left(\cos\left(\frac{1}{x}\right)\right)^{x^2}[/MATH]
Now, taking the natrual log of both sides, we can ultimately obtain:

[MATH]\ln(L)=\lim_{x\to\infty}\frac{\ln\left(\cos\left(\frac{1}{x}\right)\right)}{x^{-2}}[/MATH]
Let:

[MATH]u=\frac{1}{x}[/MATH]
And we have:

[MATH]\ln(L)=\lim_{u\to0}\frac{\ln\left(\cos\left(u\right)\right)}{u^{2}}[/MATH]
You now have the indeterminate form 0/0...Can you proceed with L'Hôpital's Rule?

 

[kanta_opu]

Thank you for replying me.

I already know how to apply L'Hôpital's Rule, so I could evaluate In(L)=-1/2.
Then we have:
L=1/sqrt(e)
Here's the right answer!

Now I understood that taking the log and using u=1/x is the key of this limit.

Again, thank you for amazing help!