Question

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

Answer 1

There is no limit.

Explanation:

The real limit of a function `f(x)`, if it exists, as `x->oo` is reached no matter how `x` increases to `oo`. For instance, no matter how `x` is increasing, the function `f(x)=1/x` tends to zero.

This is not the case with `f(x)=cos(x)`.

Let `x` increases to `oo` in one way: `x_N=2piN` and integer `N` increases to `oo`. For any `x_N` in this sequence `cos(x_N)=1`.

Let `x` increases to `oo` in another way: `x_N=pi/2+2piN` and integer `N` increases to `oo`. For any `x_N` in this sequence `cos(x_N)=0`.

So, the first sequence of values of `cos(x_N)` equals to `1` and the limit must be `1`. But the second sequence of values of `cos(x_N)` 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.