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The concept of the "limit of a function" is further generalized to the concept of topological net, while the limit of a sequence is closely related to limit and direct limit in category theory.
Main article: limit of a functionCalculus In mathematics, the limit of a function is a fundamental concept in mathematical analysis. Rather informally, to say that a function f has a limit y when x tends to a value x (or to the infinity), is to say that the values taken by the expression
Suppose f(x) is a real function and c is a real number. The expression:
means that can be made to be as close to as desired by making sufficiently close to . In that case, we say that "the limit of f(x), as x approaches c, is L". Note that this statement can be true even if . Indeed, the function f(x) need not even be defined at c.
Two examples help illustrate this concept.
Consider f(x)=x/(x2+1) as x approaches 2. In this case, f(x) is defined at 2 and equals its limit of 0.4:
| f(1.9)=0.4121 |
| f(1.99)=0.4012 |
| f(1.999)=0.4001. |
As x approaches 2, f(x) approaches 0.4 and hence we have . In the case where , f is said to be continuous at x=c. But it is not always the case. Consider
The limit of g(x) as x approaches 2 is 0.4 (just as in f(x)), but ; g is not continuous at x=2.
One need not examine limits only as x approches some finite number; one can also examine the limit of a function as x approaches infinity. For example f(x) = 2x / x+1. f(100) = 1.9802, f(1000) = 1.9980, f(10000) = 1.9998. As x becomes extremely large, f(x) approaches 2. In this case:
However, if one considers the codomainSet theory Given a function , the set B is called the codomain of f. The codomain is not to be confused with the range f(A), which is in general only a subset of B. Example Let the function f be a function on the real numbers: : defined by : The codomain of f is the extension real line, then limit of a function at infinity could be considered as a special case of limit of a function at a point.
Main article: limit of a sequenceLimit of a sequence is one of the oldest concepts in mathematical analysis. It is the essential tool in calculating pi and trigonometric functions. History The Greek philosopher Zeno of Elea is famous for formulating paradoxes that involved limiting proce
Consider the following sequence: 1.79, 1.799, 1.7999,... We could observe that the numbers are "approaching" the 1.8, the limit of the sequence.
Formally, suppose x1, x2, ... is a sequence of real numberIn mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. The term "real number" is a retronym coined in response to " imaginary number". Real numbers mays. We say that the real number L is the limit of this sequence and we write
if and only if
Intuitively, this means that eventually all elements of the sequence get as close as we want to the limit, since the absolute valueIn mathematics, the absolute value (or modulus of a number is that number without a negative sign. So, for example, 3 is the absolute value of both 3 and −3. Definition It can be defined as follows: For any real number a the absolute value of a deno |xn - L| can be interpreted as the "distance" between xn and L. Not every sequence has a limit; if it does, we call it convergent, otherwise divergent. One can show that a convergent sequence has only one limit.
The limit of a sequence and the limit of a function are closely related. On one hand, the limit of a sequence is simply the limit at infinity of a function defined on natural numberNatural number can mean either a positive integer ( 1, 2, 3, 4,. or a non-negative integer ( 0, 1, 2, 3, 4,. Natural numbers have two main purposes: they can be used for counting ("there are 3 apples on the table"), or they can be used for ordering ("thiss. On the other land, a limit of a function f at x, if it exists, is the same as the limit of the sequence xn=f(x+1/n).
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