before starting a journey, it's important to know why we're going on it. for starters, why functions? it might seem trivial, but functions are the meat of how we describe the world around us; every interaction we've ever had, everything we touch, our world as we know it is comprised of relationships between things, which is all that a function is. and in order to explore the world and then make things in that world, we're going to need to understand these functions.

so how do we get to know them? well, we've already had a bit of a taste of that from our foundational math skills, we can look at a picture to see what a function is, get values from it, and so on. but now we want to categorize these functions. we want to know their whole life stories- where they've been, what they've done, and who they're going to be. we can classify functions in tons of different ways, some of which we'll see later on in this book, but for now, we're interested in what i call the practice of divination!

in order to describe anything fully, we need to know its past, present, and future. we've already got the present: put an input into a function, get an output. but what about the past and future? divination is the art we employ to get that information about a function, through differentials and integrals. those words don't mean anything to you, but they're wyld and exciting- we're going to be constructing our own spells that allow us to divine that info just by knowing a single point and the function itself!

now, divination isn't easy, i'm sure you can see that. we've kind of got no leg to stand on right now. so let's dive into some tools we'll need to wield in order to create these spells. laying out our current tools, we've got the function itself. plug in input, get output. looking closer though, is this tool really complete? take the function 1/x. i want to know what happens at x = 0. ...well, shit. i guess we can't actually say what happens for every input.

we need an tool that we can use on any function to say "hey, what's going on here?" at any given point. since we're just having fun, we'll give it a name: the limit, since we're kind of looking for what value the function is "limited" to at that point. the limit needs to be able to take care of any nasty /0 business, and while we're at it, why don't we let it take care of plugging in wacko numbers too, like infinity! what happens to 1/x when x is infinity? i dunno, but this thing should be able to tell us.