Science Fair Project Encyclopedia
In mathematics, the concept of a "limit" is used to describe the behavior of a function, as its argument gets "close" to either some point, or infinity; or the behavior of a sequence's elements, as their index approaches infinity. Limits are used in calculus and other branches of mathematical analysis to define derivatives and continuity.
Limit of a function
Main article: limit of a function
Limit of a function at a point
Suppose f(x) is a real function and c is a real number. The expression:
means that f(x) can be made to be as close to L as desired by making x sufficiently close to c. 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 as x approaches 2. In this case, f(x) is defined at 2 and equals its limit of 0.4:
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.
A limit is formally defined as follows: Let f be a function defined on an open interval containing c (except possibly at c) and let L be a real number. The statement
means that for each there exists a such that for all x where , then .
Limit of a function at infinity
One need not examine limits only as x approaches some finite number; one can also examine the limit of a function as x approaches positive or negative infinity.
For example, consider .
- f(100) = 1.9802
- f(1000) = 1.9980
- f(10000) = 1.9998
As x becomes extremely large, f(x) approaches 2. In this case,
If one considers 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.
Limit of a sequence
Main article: limit of a sequence
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.
if and only if
- for every ε>0 there exists a natural number n0 (which will depend on ε) such that for all n>n0 we have |xn - L| < ε.
Intuitively, this means that eventually all elements of the sequence get as close as we want to the limit, since the absolute value |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 numbers. On the other hand, 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).
Main article: net (topology)
Better introduction is needed
An alternative is the concept of limit for filters on topological spaces.
Limit in category theory
Main article: limit (category theory)
An introduction will be added soon.
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