Calculus does have its limits. Indeed. In order to understand the pun of the first sentence, you need to know that calculus has two key branches: differential and integral. Although the concept of limit belongs to both branches and is an essential component to the understanding and mastery of this kind of math, differential calculus gets its name from the derivative; and such a creature depends entirely on the concept of limit. In fact, the derivative is nothing more than a special kind of limit.
Now what the heck is a limit? If you think of the ordinary definition of limit as some terminal point or boundary that is reached, then you start to get a feel for the concept of the mathematical limit. Although this concept has a formal definition, which if I stated, would probably not make a lot of sense because of the Greek letters and mathematical symbols, the actual idea is not at all hard to understand. In other words, anyone can understand the concept of limit and therefore have a good foundation toward the understanding of the calculus. In order to get this understanding, however, we must first introduce some basic definitions. These are independent variable, dependent variable, and function.
Function is one of the most important ideas in all of mathematics. In fact, most of the study of mathematics either directly or indirectly has something to do with the idea of a function. A function is nothing more than a rule, a model, a relationship between two other objects: the independent and dependent variables. The idea of function has a mathematical notation usually written as y = f(x) and read “y equals f of x.” Using this notation, y is called the dependent variable (because its value depends on what we choose for x), and x is called the independent variable.
Functions describe all kinds of things in the real world, from the growth of money at different interest rates, to the speed at which a tsunami moves in the ocean. One of the simplest of all kinds of functions is the linear function, so named because its graph produces a straight line. A linear function like y = 2x just says that whatever value we pick for the independent variable x, we get twice that value for the dependent variable y. For example, if x = 2 then y = 4, and if x = 10, then y = 20.
Now that we have laid this groundwork, we can talk in plain English–I just hate all the mathematical mumbo jumbo–of what a limit means. A limit simply means that as the value of the independent variable gets closer and closer to some value, then the dependent variable gets closer and closer to some other value. For example, if we take the linear function y = 2x just discussed, then as x gets closer and closer to the value of 2, y gets closer and closer to the value 4. At this point, you might be saying, “Okay so what’s the big deal?” Well there are times when the value of x cannot take on the limiting value, which in the example discussed was 2. In other words, we cannot calculate the value of the function when the value of x is equal to 2, yet we can still talk about what happens to the value of the function when x gets very close to the value of 2.
The idea discussed above is what gives us the concept of derivative, which is nothing more than a special limit. From the concept of derivative, all kinds of applications emerge: we can find the maximum and minimum values of functions and find rates at which one quantity is changing with respect to another at some instantaneous point. It is the derivative which permits us to find the dimensions of a rectangle if we wish to have as large an area as possible and it is the derivative that tells us the maximum height that a ball will reach when thrown according to a specific law.
Yes a simple question like “In a special relation called a function between two variables, what happens to the value of the dependent variable when the independent variable gets closer and closer to some specific value?” has spawned all kinds of mathematical discoveries by opening the door to the calculus. Not bad for a single question.