Why does $\lim_{x \rightarrow \infty} {\frac{1+\cos(x)}{1}}$ have no limit?

Question

I've tried to solve this question for a while but as of yet I can't seem to make sense of this. Why does $\lim_{x \rightarrow \infty} {\frac{1+\cos(x)}{1}}$ have no limit? Shouldn't it be $\infty$?

Answer 1

Let consider

$$x_n=2\pi n \quad x_n\to+\infty \quad n\to +\infty$$

$$y_n=\frac{\pi}{2}+2\pi n \quad y_n\to+\infty \quad n\to +\infty$$

then

$${\frac{1+\cos(x_n)}{1}}\to2\neq{\frac{1+\cos(y_n)}{1}}\to1$$

thus

$$\lim_{x \rightarrow \infty} {\frac{1+\cos(x)}{1}}$$

does not exists.

Answer 2

For some limit as $x\to x_0$ to be equal to $\infty$, this means that we can make the function arbitrarily large as we get closer to $x_0$. In this example, the function ($1+\cos x$) never gets larger than $2$. So the limit cannot be $\infty$.

By considering two different sequences, one where $\cos x=1$, and one where $\cos x=-1$, you can show that along two different paths as $x\to\infty$, we get two different limits. So this means that no limit exists.

Answer 3

$$1+\cos x = 2 \cos^2 \left(\frac{x}{2}\right).$$

$\cos x$ oscillates between $1$ and $0$ in the first quadrant even as $n \to \infty$. Hence $\cos \frac{x}{2}$ also oscillates. I.e. it does not tend to a definite limiting value.