Proof Verification: $\lim_{x \to 0} \cos(1/x) $ does not converge

Question

This is one of the questions from Bartle. While attempting this question, I used the sequential criteria as follows:

take $\;(x_n)=1/n$ Then, as $ n \to \infty$ $(x_n) \to 0$

Now, $\cos(1/x) = \cos (1/1/n) = \cos(n)$ diverges as it oscillates between -1 and 1

Can anyone please verify if this approach is correct? The reason why I'm a bit doubtful is because the solution manual uses a different approach that makes use of $\pi$ and what not and was worried if this rather simplistic approach is ok?

Thank you in advance.

Answer 1

To complete we need also to prove that $\cos n$ oscillates.

As an alternative let consider as $n\in \mathbb{Z}$ with $n\to \infty$

• $x_n=\frac1{2\pi n}\to 0\implies \cos \frac1{x_n}=\cos 2\pi n=1$
• $x_n=\frac1{\pi (2n+1)}\to 0\implies \cos \frac1{x_n}=\cos ((2n+1)\pi)=-1$

thus the limit doesn't exist since we have two subsequences with different limits.