I want to prove if sin(1/x) diverges or converges as x approaches infinity. I know the answer is diverge because the function oscillates. I want to prove it more rigorously though, for my own well being and for the exam coming up. I am having trouble with the ratio test and I rewrote the expression as sin(1/x)[sin^2(1/x) +cos^2(1/x)] to see if it would help but that won't because if I set b equal to sin^2(1/x) +cos^2(1/x) that is not smaller than the original expression of course so I can't use the direct comparison test. I assume I will be using the limit comparison test but I need a little kick start. Thanks.