one of the most core questions we can ask about a function is "what is it doing?" functions do different things in different ways, and we'd like to be able to characterize the differences between them.

the simplest way we can do that is to look at points on the function. that might be a specific point (like where there's a hole or a particularly sharp spot), or maybe it's comparing two different points. when we compare two different points, we tend to also include information about the inputs of those points. this might seem familiar to you from your basic understanding of functions- say, if we wanted to find the Slope of a function, we would need the values we put in, as well as the output values.

it turns out that slope is actually quite a useful object. when a function changes, it changes a certain amount over a certain period. this can be thought of as the "speed" of the function. and when we're in a space with multiple basis vectors (r2 and onwards), the function is also changing in a certain direction- it can turn someway instead of always being on a line. combining the speed with the direction gives us a vector, "velocity".

now, all of this is probably familiar, but it motivates why we care about velocity in the first place- it's not just a physics concept, its an important quantity of a function. if you were walking along the countryside, it would be quite a different experience if you were strolling or sprinting! so we're interested in finding out what these velocities are for any function, and actually, we're interested in the rate of change OF the velocity as well: how fast are we getting faster?

let's look at a picture and try to figure out that change! we're gonna chill out in r2 for now, since we don't really need to worry about other dimensions. if we can get it to work here, it should work wherever we'd like to be. to find the slope of this function, we can arbitrarly pick two points out and draw a line between them. the slope of that line will be the change in y over the change in x! now, since we want to write it a bit more generally (and we're interested in information about f), we swap those y components over to f(x).

then we can do a little funny trick. what if, instead of saying point 1 and point 2, we defined everything around one point? it would be kind of nice if we could just pick one point and not have to compute another one, and then we can just plug in a distance about how far away the other should be. this way, if we want a smaller line, we can plug in a smaller value. well, this is pretty neat! now we've got a formula to draw a line between two points on a function and give us a slope, and we only have to put one point in to get it.

alright, well you can get mad at me a little if you want. i told you we needed to find the speed and we just went off and drew some lines on a function. not very polite of me. let's break down how we might get a speed... i guess we know how to get the speed if the function was just a line, right? that would straight up be the slope! but that funny guy up there isn't a line at all. humm. well, is there a way we could make it a line? i mean, we could stretch it out.. but that doesn't really help us.

woah holy shit when i looked at it really close, it kind of looked like a line... if we just zoomed in REALLY FAR, you know, i bet any function could look like a line...! our formula from last time should work for this- we just need to put a really really small value for that difference between the points and i bet it would work! but we can only put a distance in that's so small.. hm. like maybe 0.0000000000000001 or something. but if we want to be PERFECT it should be 0. urgh, but that breaks our function...

re-enter: our limit! this is the perfect application for it, we've got a very scary place looming over us that we want to discover, so let's unpack our spell kit and get it all set up.