Limit as $x$ approaches infinity of $\cos(\pi x)$

Question

This question is if $\cos(\pi n)$ and $\sin(\pi n)$ have a limit and how to calculate it. Do $\lim_{n\rightarrow \infty} (\cos(\pi n)-\frac{1}{n})$ and $\lim_{n\rightarrow \infty} (\sin(\pi n)-\frac{1}{n})$ exist ? If yes how to calculate them?

Answer 1

You could consider the two subsequences $\cos(2k\pi)$ and $\cos((2k-2)\pi)$ where $k$ is a natural number. Do something similar for $\sin(\pi n)$. Notice that the two subsequences have distinct limits so clearly the sequence doesn't converge.