Given:
I understand that sin(x) oscillates between -1 and 1 as we go towards infinity in either direction, but if I didn't know what the graph looked like or the general behavior of sin(x), how would I write algebraically to show that the limit doesn't exist? The work I show is usually plugging in large numbers close to each other to show that the y-values of sin(x) oscillate, but I don't think that is a valid way of doing this unless I use like 10 different x-values and plug them all in.
Intuitively, if you are evaluating a limit as x approaches infinity, you are making x approach a very large value. So if there was a limit, then if you choose a very large value, say "d", then by making x greater than this, you should be guaranteed to have the difference between f(x) and L be within your tolerance, e.