Calculus comes from Latin for "small stone." It deals with functions and what happens when you break them into very small parts, infinitesimally small. There are two main branches of calculus: differential calculus and integral calculus. Differential calculus deals with derivatives, and integral calculus deals with integrals. The first thing you will need to know is how to evaluate limits for functions.
Imagine a function f(x), and some value of x, c. Now, imagine that you evaluate the function for values that are extremely close to c, but not equal to c. You could evaluate it for c - 1, then c - 0.5, then c - 0.1, then c - 0.000001, and so on. You could also evaluate it for values like c + 1, c + 0.5, c + 0.1, c + 0.000001, and so on, getting closer and closer to c. The value that f(x) approaches when x approaches c is called the limit of f(x) as x approaches c. Common sense might tell you that the limit of f(x) as x approaches c is simply f(c). However, for some functions, that might not be the case.
For example, let us evaluate the limit of 1/x2 as x approaches 0. Here, f(x) is 1/x2 and c is 0. Let us try plugging in some values for x that are close to 0. If x is 1, f(x) is 1. If x is 1/10, f(x) = 1/(1/10)2 = 1/(1/100), which is 100. If x is 1/100, then f(x) is 1/(1/100)2 = 1/(1/10000), which is 10000. We can see that as x gets closer and closer to 0, f(x) increases boundlessly. If we approach from the other side, when x is less than 0, we get the same result. f(-1) = 1, f(-1/10) = 100, f(-1/100) = 10000. Thus, we can say that the limit as x approaches 0 of 1/x2 is infinity. This is one of the cases where the limit of a function at a certain value for x is not the same as the value of the function there, because the function is not defined at x = 0.
Sometimes the limit of a function at a certain place doesnt exist. For example, let us try to find the limit of 1/x as x approaches 0. It might be tempting to say that the limit is infinity, because if you plug in positive values for x that are close to 0, the function approaches infinity. However, if you plug in negative values that are close to 0, the function will approach negative infinity. Since the limits from the positive and negative sides don't match, we say that the limit doesn't exist.The notation of limits is as follows:
Evaluate these following limits:
Now that you know the basics of limits, you can move on to learning about derivatives. It is best to start with derivatives, then go to integrals.