the past and the future are complicated beasts, so let's focus on the present. before venturing out, we need to examine the one tool we're already familiar with: using our map. currently, if we want to know where a function is at any given point, we simply input the point into our map and see where the output goes.

let's look at an example, here's a little friend 1/x! if we put x = 5, we get 1/5. seems pretty good! we know that we define a domain which holds the inputs that we're allowed to plug into the function. in this case, our domain wouldn't include 0 because there's an asymptote at 0. if we try to input 0, the function gets really dizzy and throws up, which is fun for nobody.

the trouble is, i'm greedy, and function analysis is greedy. i need to be able to plug in anything to our function and see what happens without the world blowing up. there's some behavior that happens at 0, for sure, cause i can see it! it goes up really high, so high we can't see it! actually, before we get into building this tool, let's talk about that place for a second.

lots of our functions aren't tame. they (like us) have rich and fulfilling lives they get up to, and that includes .. unknowable behaviors, so to speak. in general, our inputs and outputs can go to infinity and negative infinity! we can't visual infinity, so as long as something gets really big forever, we say that it's going to infinity (and if that number is negative, it would go to negative infinity). they might go to those places quickly or slowly, but we just need to be comfortable with the fact that they're going to go there!

so now we've got our goal: a tool that we can apply to any function, which lets us throw ANY number into it and gives us information about what the function is doing there. first, it needs a name. since we want to expand our understanding of the function to eeeeverywhere the function is, we're kind of testing its limits... let's call this operation the limit! when we want to take the limit of a function, we'll write it like this! [limit picture 1]. now we need to be able to plug a number into it, and just to make sure this will work for when we have multiple inputs, we'll specify what variable we're doing. [limit picture 2]

now let's define some properties of our limit and how it works. for our simplest case, if we plug a number in our domain into the limit, it'll spit out the answer we expect! nice.