# Multi variable Limit to $\infty$

The question says that if $$\frac{\cos x}{\sin ax}$$ is a periodic function then find the find the value of $$\lim _{m \rightarrow \infty} \lim_{n \rightarrow \infty} \left(1+ \cos^{2m} n! \; \pi a\right)$$ I really don't have any clue on how to begin the question. I have never seen a question like this. The only thing I could think of since is that since $$\frac{\cos x}{\sin ax}$$ is periodic then we can assume $$a$$ to be $$1$$ since $$\cot x$$ is periodic. So then we have to find the limit of the function, $$\lim _{m \rightarrow \infty} \lim_{n \rightarrow \infty} \left(1+ \cos^{2m} n! \; \pi \right)$$ But even this is based on an assumption and I have no idea on how to go from here. Any clue or hint would be much appreciated.

## 1 Answer

... we can assume $$a$$ to be 1...

We can (must) assume $$a$$ to be rational.

In your case (the general case is almost the same) the inner limit $$\lim_{n\rightarrow \infty}\left(1+\cos^{2m}n!\pi\right)$$ is trivial because for $$k\in\Bbb Z$$, $$\cos(k\pi) = \cdots$$

• Alright so we can just say that since anything of the form $\cos kx = 1$ then we can say the limit of the function is $1+1=2$? Jan 28 '19 at 14:39
• I did actually think of the problem in that way but I just wasn't convinced by that argument Jan 28 '19 at 14:39
• @PrakharNagpal, yes. In the general case, $n!a$ is integer for $n$ large enough. Jan 28 '19 at 14:41
• Alright, thank you so much! Jan 28 '19 at 14:42
• @PrakharNagpal, $\cos k\pi = \pm 1$. Jan 28 '19 at 14:42