How do I prove that $\lim_{x\to \infty} |\cos x|^x$ doesn't exist?

Question

How do I prove that $\lim_{x\to \infty} |\cos x|^x$ doesn't exist?

I know that it doesn't because I have seen the graph , but is there a way I can proof using algebra ?

Answer 1

Guide:

• Consider the sequence $x_n = 2n\pi$ and another sequence $y_n = 2n\pi+\frac{\pi}2$. show that the function value of $f(x)=|\cos x|^x$ evaluated at these sequence values go to different limit.

Answer 2

Using Heine theorem.

Take $x_n=2n\pi$ and $x'_n=2n\pi+\frac{\pi}{3}$, both are tending to $\infty$. But $f(x_n)=1\to 1$ and $f(x'_n)=(1/2)^{2n\pi+\frac{\pi}{3}}\to 0.$

Answer 3

The range of the function keeps being $[0,1]$ on every period of the cosine.