left me behind at "decreasing in amplitude by some amount"

The function y=sin(x) is a sine wave with an amplitude of 1. If you have the function y=sin(x)/x it will be undefined at x=0 but beyond that it will be a sine wave of decreasing amplitude. At x=2 it will be a sine wave of amplitude 1/2 and at x=3 it will be amplitude 1/3. This sine wave would be said to be asymptotic to the y-axis since its amplitude would be approaching 0 as x increases without bound. It will never reach 0 because, for any finite x, 1/x>0.