# What is the limit as x approaches infinity of cosx?

Jul 14, 2015

There is no limit.

#### Explanation:

The real limit of a function $f \left(x\right)$, if it exists, as $x \to \infty$ is reached no matter how $x$ increases to $\infty$. For instance, no matter how $x$ is increasing, the function $f \left(x\right) = \frac{1}{x}$ tends to zero.

This is not the case with $f \left(x\right) = \cos \left(x\right)$.

Let $x$ increases to $\infty$ in one way: ${x}_{N} = 2 \pi N$ and integer $N$ increases to $\infty$. For any ${x}_{N}$ in this sequence $\cos \left({x}_{N}\right) = 1$.

Let $x$ increases to $\infty$ in another way: ${x}_{N} = \frac{\pi}{2} + 2 \pi N$ and integer $N$ increases to $\infty$. For any ${x}_{N}$ in this sequence $\cos \left({x}_{N}\right) = 0$.

So, the first sequence of values of $\cos \left({x}_{N}\right)$ equals to $1$ and the limit must be $1$. But the second sequence of values of $\cos \left({x}_{N}\right)$ equals to $0$, so the limit must be $0$.
But the limit cannot be simultaneously equal to two distinct numbers. Therefore, there is no limit.