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Why Study Calculus – The Limit

Calculation has its limits. Indeed. To understand the pun in 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 for the understanding and mastery of this type of mathematics, the 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 type of limit.

Now what the hell is a limit? If you think of the ordinary definition of limit as a terminal point or limit that is reached, you begin to understand the concept of a mathematical limit. Although this concept has a formal definition, which if I were to say it, probably wouldn’t make much sense because of the Greek letters and mathematical symbols, the actual idea is not at all difficult to understand. In other words, anyone can understand the concept of limit and therefore have a good foundation for understanding calculus. In order to understand this, 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, directly or indirectly, has something to do with the idea of ​​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 that is usually written as y = f(x) and read “y equals af of x”. Using this notation, y is called the dependent variable (because its value depends on what we choose for ax), ix is ​​called the independent variable.

Functions describe all sorts 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 functions of all types 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 choose 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’ve laid that groundwork, we can talk in plain English (I just hate all the math crap) about 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 another value. For example, if we take the linear function y = 2x just discussed, then as x gets closer and closer to the value 2, y gets closer and closer to the value 4. At this point, you might be saying : “Okay then. what’s the big deal?” Well, there are times when the value of x cannot take the limiting value, which in the discussed example was 2. In other words, we cannot calculate the value of the function when the value of x is equal to 2, but 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. All kinds of applications arise from the concept of derivative: we can find the maximum and minimum values ​​of functions and find the rates at which one quantity is changing with respect to another at some instantaneous time. It is the derivative that allows us to find the dimensions of a rectangle if we want to have an area as large as possible and it is the derivative that tells us the maximum height that a ball will reach when launched according to a specific law.

Yes, a simple question like “In a special relationship called a function between two variables, what happens to the value of the dependent variable as the independent variable gets closer and closer to a specific value?” it has generated all kinds of mathematical discoveries opening the door to calculation. Not bad for a single question.

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