Extension: Limits of Ratios f(x)/g(x), by Order of Growth

Calculating limits by knowing the ``order of growth'' of various terms is an extremely fast and useful method for finding limits.

First, we'll look at end behavior and order of growth, that is, limits as $x\to\pm\infty$.

One way to compute this limit is by repeated applications of L'Hopital's rule.

TODO: other proofs

But this is going to be such a useful result that we don't want to have to use L'Hopital's rule every time we see a limit like this. Instead, we'll learn some key facts about orders of growth. Here, the key fact is:

$e^x$ grows faster than any power of $x$. In fact, $e^x$ is said to ``dominate'' $x^k$, that is, $\lim_{x\to+\infty}\frac{x^k}{e^x}=0$.

Intuitively, this should make sense. Recalling the Maclaurin expansion of $e^x=1+x+\frac{x^2}{2!}+\cdots$ and the fact that polynomials of higher degree dominate polynomials of lower degree, it makes sense that eventually $e^x$ will dominate any (finite-degree) polynomial.

This limit can also give us some other useful facts:

$x^k$ grows faster than $\log x$. TODO justify

TODO examples

TODO limits towards zero

As $x\to0^+$, $\frac{1}{x}$ blows up faster than $\log x$. $$\lim_{x\to0^+}x\log x=\lim_{x\to0^+}\frac{\log x}{1/x}=0.$$ We can get this result by making the substitution $t=\frac{1}{x}$, to get $$\lim_{x\to0^+}x\log x=\lim_{t\to+\infty}\frac{-\log t}{t}=0$$ by one of the earlier key facts.

TODO: Big-O notation