Question

Evaluate the following: $\underset{n\to \infty }{lim}\frac{\cos n+\sin n}{n^2}$,

Answer 1

We have,

$0\le \left|\frac{\cos n+\sin n}{n^2}\right|\le \frac{\sqrt{2}}{n^2}\forall n\in N$......(1) [ Since $|\sin n|\le 1$ and $|\cos n|\le 1$ for $n$ being natural number]

Now from (1) it is clear as $n\to \infty$, $\frac{sinn+cosn}{n^2}\to 0$.

So,

$\underset{n\to \infty }{lim}\frac{\cos n+\sin n}{n^2}=0.$