lim as x approaches infinity of x to the power of x.

or alternately

lim as x approaches 0 of 9999/x

Done. The second one goes to infinity* faster, of course.

I’m looking at all the answers with the fastest growing functions and I’m impressed. But I am too lazy to go Googling ways to generate even faster growing functions, so let me enjoy my tweaks with logic and head in the opposite direction. The truly fastest growing curve is the one that has a vertical asymptote**, so just set up a limit and voila.

Other functions with vertical asymptotes: [math]tan(x)[/math], [math]csc(x)[/math], [math]sec(x)[/math]. Those are just some I can think of right now; I’m sure there are others.

*Yeah, yeah, yeah. Infinity technically isn’t a number. To counteract this, just use a table and input a really tiny number for x. Say, [math]10^G[/math], where G = negative Graham’s number. Try that out on paper, and you get [math]9999*10^g[/math], where g = positive Graham’s number. If I wanted to be even more excessive, I would replace 9999 with something ridiculous like [math]tree(g!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!)[/math].

**If anyone is unfamiliar, an asymptote is a numerical value of x or y that a function never touches/crosses (there are exceptions with horizontal asymptotes, but not vertical asymptotes). If it’s a vertical asymptote, the function goes to infinity at any x-value arbitrarily close to the asymptote. If it’s a horizontal asymptote, the function never equals the y-value of the asymptote, either getting super close to it as x goes to infinity or wavering around it (i.e [math]1/sin(x)[/math]).

Lesson of the day: Inverse functions give you unlimited power.

EDIT: I don’t know why I keep editing this. I never seem to be able to get all my thoughts down in one go. Probably because I should be working on my paper right now.