Epsilon-N proof

Question

Hello I want to prove that

$$\frac{1+\sin n}{n^2+1}\rightarrow0 \text{ as } n\rightarrow\infty$$

Using the Epsilon-N proof. I'm not sure what to do during the scratchwork. How do I work with the $\sin n$ ?

Answer 1

We can consider choosing $n$ to be large enough that $$\left|\frac{1+\sin n}{n^2+1}\right|\le \frac{2}{n^2}< \epsilon$$

Try to solve for how big $n$ should be for $\frac2{n^2} < \epsilon$ to hold and hence we can bound our original function.