Believe it or not, the simple, humble limit is the most important thing in calculus, by far. Most people fly over the limit, on their way to limits, integrals, and sequences & series. Still, without the simple limit, absolutely none of that would be possible.
The Introduction of Limits
Limits allow us to do things that algebra, our best tool before calculus, explicitly forbids. They give us such a great wealth of power that in reality, calculus is defined by the things that we can do when empowered by the limit.
For most of our history, algebra was limited by a few problems. In the end, the biggest thing holding us back in our study of physics was our complete inability to divide 0 by 0. It sounds silly, but “0/0″ can be any number of things. It should be infinity, or it should be a number, but as far as any mathematics of the day, there was absolutely no way to tell.
Newtonand Leibniz both had the insight, more or less simultaneously, that they could still work with moments where 0/0 was approximately a number as you got close. For instance, suppose the problem is (x-2) / (x-2). For any number except two, the answer is obviously 1. At 2, we get 0/0 which does not exist. Newtonand Leibniz both decided that in clear cut cases where the number that is missing could just be filled in with obvious reasoning, they should just treat it as the “correct answer.” They said that as you get close enough to a number, called a in this article, on the x-axis, your y-value gets closer and closer to the limit, L.
The Formal Limit
Formally, the mathematics of the day was considered pretty loosey-goosey. In fact, it would never fly in a modern setting, so a “formal definition” for a limit was created. Called the “epsilon-delta definition,” it formalizes the process of creating a limit. The true definition is very hard to relate, but I’ll summarize it as an easy process.
Imagine you have an evil opponent who challenges your limit. You say the limit answer is a number, and you call it L. To challenge you, he gets to name a (typically small) number called epsilon. He says you can’t guarantee that you can get that close.
You fight back with some algebra. Using typically algebraic concepts, you shoot back another number, called delta. Your delta says that as long as you are within a small gap of the desired a (a gap no bigger than delta), you are no further than epsilon from the limit, L.
Since your opponent can name any number, infinitely small, for epsilon, you will squeeze closer and closer to the limit in delta, and if you can do it for any number, you win and your limit is correct.
Using Limits
From that point on, you are empowered with this nifty tool, the limit. Typically, you won’t go through the epsilon-delta process for a normal limit, but the point is that you could, and that is all that matters. Now, with your new powers, 0/0 becomes an answer… sometimes. When it does, you can get more mathematics.
Calculus creations in general work as follows. You create an estimate for the thing you want. You find a way to refine it to make it closer and closer to perfect. Because you get a 0/0 type error at the “perfect” answer, you take a limit. Voila, you have the perfect answer, care of the limit expanding your algebra powers into the realm of calculus.
Final Notes
Calculus isn’t actually very hard. The hardest part of most calculus is knowing what algebra you need to do. Typically, in a given problem there is only one step that is really calculus, and it is usually quite straightforward once your algebra is out of the way. Calculus, defined as algebra with limits, suddenly becomes a very useful tool in the real world when dealing with things that are changing.
Sources:
Wolfram Math World: Limit