# Is my analysis of this limit reasonable?

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)?

I think you’re on the right track, but your exposition is bad. Don’t ever write 1/infinity if you want to be taken seriously. The right way to do this, assuming A[sub]0[/sub] > 0, is to note that

and apply the squeeze theorem.

Edit: Have you taken real analysis? This is a bread and butter sort of problem there, and if you want to even read about the theoretical side of machine learning, you need to be pretty comfortable with it.

It’s been a long time, 3 years actually, since I’ve done continuous/real analysis. I do discrete/integer and probability analysis more often.

ETA: I’m more trying to refresh my memory, my program never uses calc or continuous analysis despite requiring it, so it’s atrophied a tad. Even the machine learning courses focus on probability calculus a lot more than it uses reals.

And I know 1/(infinity) is evil (all my math teachers would mark off for it), but typing math on vBulletin is annoying.

The issue isn’t the typing. It’s that 1/infinity is a meaningless sequence of symbols, because infinity is not a number. I can’t stress this strongly enough: any approach that relies on plugging in infinity like it is a number is fundamentally flawed. You have to honestly engage the notion of a limit at infinity and work rigorously.

If you’re doing anything even remotely applied, you will very likely be expected to read Boyd & Vandenberghe at some point. Take a look at the appendix on the mathematical background, and see what you have to do in order to be familiar with what they’re talking about.

Other than using different notation and terminology than I’m used to, I already knew everything in that appendix, and that which I didn’t (which was mostly the schur complement) was easily understandable once I figured out their terminology.

At least, not in this context.