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[tex]\lim_{x\rightarrow ∞}\frac{2^x}{x^2} = ∞.[/tex]

We "know" this because 2

^{x}grows faster than x

^{2}. I use quotes because this is just what we're told in basic calculus classes. But what about a theorem for this? I've searched through google, looked through various university homework pages, and in all of them I couldn't find any standardized way to determine if one function grows faster than another; no theorem describing this behavior precisely and why.

Obviously this has been covered somewhere, and I just don't know how to find it. Earlier I was trying to find out information about the series (-1)

^{n}, and could find very little until I stumbled upon the phrase "Grandi's series." I'm hoping there is something out there describing this issue as well, so I can get some more precise knowledge. Simply looking at the function on my graphing calculator feels insufficient.

So in summary, I'm hoping someone can direct me to a more rigorous and precise set of definitions regarding limits of f(x)/g(x) with x approaching infinity where the relative growth rates of f(x) and g(x) are important in determining that limit. Specifically, why one type of function grows faster than another, and what the criteria are that determine that.

Any insight into this would be very appreciated!