This graph, which charts the function f(x)=1x, shows that the value of the function approaches zero as x becomes larger and larger. Yet even as x approaches infinity, the value of the function will never quite fall to zero. Zero, therefore, is said to be the limit of this function.
Calculus (mathematics), branch of mathematics concerned with the study of such concepts as the rate of change of one variable quantity with respect to another, the slope of a curve at a prescribed point, the computation of the maximum and minimum values of functions, and the calculation of the area bounded by curves. Evolved from algebra, arithmetic, and geometry, it is the basis of that part of mathematics called analysis.
Calculus is widely employed in the physical, biological, and social sciences. It is used, for example, in the physical sciences to study the speed of a falling body, the rates of change in a chemical reaction, or the rate of decay of a radioactive material. In the biological sciences a problem such as the rate of growth of a colony of bacteria as a function of time is easily solved using calculus. In the social sciences calculus is widely used in the study of statistics and probability.
Calculus can be applied to many problems involving the notion of extreme amounts, such as the fastest, the most, the slowest, or the least. These maximum or minimum amounts may be described as values for which a certain rate of change (increase or decrease) is zero. By using calculus it is possible to determine how high a projectile will go by finding the point at which its change of altitude with respect to time, that is, its velocity, is equal to zero. Many general principles governing the behavior of physical processes are formulated almost invariably in terms of rates of change. It is also possible, through the insights provided by the methods of calculus, to resolve such problems in logic as the famous paradoxes posed by the Greek philosopher Zeno.
The fundamental concept of calculus, which distinguishes it from other branches of mathematics and is the source from which all its theory and applications are developed, is the theory of limits of functions of variables (see Function).
Let f be a function of the real variable x, which is denoted f(x), defined on some set of real numbers surrounding the number x0. It is not required that the function be defined at the point x0 itself. Let L be a real number. The expression
is read: “The limit of the function f(x), as x approaches x0, is equal to the number L.” The notation is designed to convey the idea that f(x) can be made as “close” to L as desired simply by choosing an x sufficiently close to x0. For example, if the function f(x) is defined as f(x) = x2 + 3x + 2, and if x0 = 3, then from the definition above it is true that
This is because, as x approaches 3 in value, x2 approaches 9, 3x approaches 9, and 2 does not change, so their sum approaches 9 + 9 + 2, or 20.
Another type of limit important in the study of calculus can be illustrated as follows. Let the domain of a function f(x) include all of the numbers greater than some fixed number m. L is said to be the limit of the function f(x) as x becomes positively infinite, if, corresponding to a given positive number e, no matter how small, there exists a number M such that the numerical difference between f(x) and L (the absolute value |f(x) - L|) is less than e whenever x is greater than M. In this case the limit is written as
For example, the function f(x) = 1/x approaches the number 0 as x becomes positively infinite.
It is important to note that a limit, as just presented, is a two-way, or bilateral, concept: A dependent variable approaches a limit as an independent variable approaches a number or becomes infinite. The limit concept can be extended to a variable that is dependent on several independent variables. The statement “u is an infinitesimal” meaning “u is a variable approaching 0 as a limit,” found in a few present-day and in many older texts on calculus, is confusing and should be avoided. Further, it is essential to distinguish between the limit of f(x) as x approaches x0 and the value of f(x) when x is x0, that is, the correspondent of x0. For example, if f(x) = sin x/x, then
however, no value of f(x) corresponding to x = 0 exists, because division by 0 is undefined in mathematics.
The two branches into which elementary calculus is usually divided are differential calculus, based on the consideration of the limit of a certain ratio, and integral calculus, based on the consideration of the limit of a certain sum.