Limits are usually one of the first topics studied in a calculus class. An easy example of a limit is the sequence 1/2, 1/3, 1/4 and so on forever. It is fairly obvious that the "limit" of this sequence is zero. It is not part of the sequence, but you can get as close to zero as you want if you go far enough in the sequence. Some limits are not so obvious. The expression N^2/(N^2 + 2N + 3), as N goes from one to infinity, starts 1/6, 4/11, 9/18 and so on. For most people it is not obvious that the limit is one. The limit function of series and sequences are used to define many of the other functions used in calculus.
The derivative of a function describes how the function changes at each point. If D is the derivative of a polynomial P, plugging a point into D will give the slope of the line that is tangent to the curve at that point. For example, the polynomial Y = X^2 is a parabola that goes through points (1,1) and (2,4). The derivative of X^2 is 2X, so the slope of the tangent line at the point (1,1) is 2X = 2(1) = 2, and the tangent line that goes through (1,1) is Y = 2X - 1. Similarly, the tangent line at the point (2,4) is Y = 4X - 4.
The fundamental law of calculus starts that derivatives and integrals are inverses of each other. If a function F is graphed, the area between the curve and the X axis is given by the integral of F. Using the calculus integral function, you can find the area of anything you have an equation to describe. Using double integrals, you can find the volume of three-dimensional objects.
Many exponentials and logarithms --- the inverse of exponentials --- have a central role in calculus functions. For example, they are often part of the solutions to differential equations --- equations that have derivatives in them. One of these is especially frequent: e^X where "e" is Euler's number. One of the interesting relationships involving e^X is the fact that e^X = the derivative of e^X = the integral of e^X. Another interesting relationship is the fact that the area under the curve Y = 1/X between the Y axis and a point "p" is equal to the logarithm of "p" to the base "e."