For fun and to help myself understand it better, I was trying to show myself what the pieces of a simple dampened harmonic oscillation function do.

A[sub]0[/sub] e[sup]-bt[/sup]cos(ωt + φ), to prove to myself that e[sup]-bt[/sup] was the dampening term (in other words, the term that makes it degrade over time). I took the limit as t goes to infinity (while assuming b > 0) to show that it causes it to have an amplitude of 0 (among other things, like taking limits of b).

I know the result I get is correct, but I’m not sure my method is mathematically sound.

Cosine doesn’t converge, which was causing me an issue, but it has a period, so I decided to do this

lim t->infinity (e[sup]-bt[/sup]) = 1/(infinity)

A[sub]0[/sub]/(infinity) = approx = 1/(infinity)

1/(infinity) * cos(infinity)

Then I said, okay, cosine is bound at [-1,1], so I said

1/(infinity) * -1 = 0

1/(infinity) * 1 = 0

So overall the range of the function is [0,0] as t goes to infinity, meaning the function overall goes to 0.

Again, I’m almost certain I’m correct in the final number, but that doesn’t mean the method is sound, I’ve done mathematical proofs before, I know one case, or even many cases, can mask that a method doesn’t really work. So is my intuition based on really shaky mathematical ground, or does it work correctly (even if not ideally)?