Question

In the following problems, find the limit of the given sequence as n approaches infinity. 9. n sin (1/n).

Answer 1

You get a 0*infinity case so you need to use l'hospital's. So do this:
\[\lim_{n \rightarrow \infty} \frac{\frac{d}{dn}\sin(1/n)}{\frac{d}{dn} \left(\frac{1}{n}\right)}=\lim_{n \rightarrow \infty}\frac{\frac{-1}{n^2}\cos(\frac{1}{n})}{\frac{-1}{n^2}}=\lim_{n \rightarrow \infty}\cos(\frac{1}{n})=1\]