Convergence of nsin(1/n)? (10 pts best answer)

My textbook says that the sequence nsin(1/n) converges to 1. However, I don't understand how that works at all. I see that you need to rewrite this sequence as sin(1/n) / (1/n), then as the limit of n reaches infinity, this will become 0/0. I am stuck here, as l'Hopital's rule wouldn't get me anywhere either.

Thanks!

Let x = 1/n, and note that as n approaches infinity, x approaches 0.

So sin(1/n) / (1/n) = sin x / x, which approaches 1 as x approaches 0 from a standard fact in calculus.
Therefore, nsin(1/n) converges to 1 as n approaches infinity.

(If you did not know this standard fact, then you could use L'Hopital's rule to find
lim x -> 0 of sin x / x:

lim x -> 0 of sin x / x
= lim x -> 0 of cos x / 1
= cos 0 / 1
= 1.)

Lord bless you today!

Are you absolutely sure that l'Hopital's rule will get you nowhere?