# How to properly express $\lim\limits_{x \to \pm\infty}$ with sine and cosine

Often, I'm asked to solve limits for expressions that contain sine or cosine. Limits for sine or cosine do not exist in $\infty$. But yet, limits with sine and cosine may have a solution:

$$\lim\limits_{x \to \infty} \frac{x+\sin(x)}{x+\cos(x)} = \lim\limits_{x \to \infty} \frac{\infty+???}{\infty+???} = 1$$

I wonder how should I properly write this and what is used instead of question marks. I felt like using $<-1,1>$ here, but that's a funny solution.

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Divide top and bottom of your expression by $x$ and apply known limits and limit rules. (And don't treat "$\infty$" as if it were a number.) –  David Mitra Jan 21 '14 at 15:57
Your procedure is incorrect: you cannot take limits of individual terms and combine them. Instead, think what happens when you plug in a large number for $x$, and then an even larger one. –  user66081 Jan 21 '14 at 15:58
@DavidMitra And what $\frac{sin(\infty)}{\infty}$ should be? It sounds like a $0$ to me. That would make my example $\frac{0}{0}$ then... –  Tomáš Zato Jan 21 '14 at 16:10
Ah damn, forgotten the $1$. So it's $\frac{1+0}{1+0}$. I get it now. –  Tomáš Zato Jan 21 '14 at 16:11
If I understood well how it works, @egreg, I'd never post this question. –  Tomáš Zato Jan 21 '14 at 16:51