Redirected from Converge (topology)

Suppose f(x) is a real function and c is a real number. If the values of f(x) approach (get close to, but don't necessarily reach) the number L, as x approaches c, one can state that "the limit of f(x), as x approaches c, is L" and write
If the values of f(x) approach infinity as x approaches c, one writes
In "wellbehaved" functions (i.e. in continuous ones), the limit, as x approaches c, can be found by directly substituting c for x. For example, if f(x) = 7; as x approaches 32, the limit is 7 (the limit of a constant is a constant). Another example is f(x) = 2x  5; in that situation, as x approaches 3, f(x) approaches f(3) = 2·3  5 = 1.
Limits are more interesting when they are unreachable; for example: if f(x) = (x³  1) / (x  1) then, x cannot equal 1 (as that would result in division by zero); however, f(x) does approach some number c (as x approaches 1). f(0.9) = 2.71, f(0.99) = 2.9701, f(0.999) = 2.997001, f(1.1) = 3.31, f(1.01) = 3.0301, f(1.001) = 3.003001  We see that, as x approaches 1, f(x) approaches 3; however, x never equals 1 and f(x) never equals 3. The limit can be verified using algebra; since: (x^{2} + x + 1)(x  1) = (x^{3}  1)...if g(x) = x^{2} + x + 1, g(x) = f(x) (for any x ≠ 1). g(x) and f(x) are identical at every point, except 1; f(x) has a hole at 1. Via direct substitution (x = 1) one can determine the limit of g(x) to be 1^{2} + 1 + 1 = 3; which is the value f(x) approaches, as it approaches the hole.
Here we have used the following rule: if it so happens that if g(x) = f(x) for all values of x, except for c; then, so long as g(x) has a limit, that limit will be equal to the limit of f(x).
The above also shows that whether or not f(c) exists has no bearing on whether or not the limit of f(x) (as x approaches c) exists. If f(c) exists, then its value has also no bearing on the limit.
Not every function has a limit at every point. Consider:
When trying to evaluation a limit by simply substitution c into the function, one often would have to divide by zero, which is of course impossible. Here l'Hôpital's rule helps: If f(c)=0, g(c)=0, and both derivatives f'(c) and g'(c) are defined, then:
If three functions f(x), g(x) and h(x) are given such that h(x) ≤ f(x) ≤ g(x) and if the limit, L, (as x approaches c) of h(x) is equal to the limit (as x approaches c) of g(x); then, the limit of f(x) not only exists, but is also equal to L. The function f(x) is "squeezed" between g(x) and h(x).
One might ask whether there is some relationship between f(x)  L (the absolute value [of the function, at x, minus its limit, L, as x approaches c]) and x  c (the absolute value [of x minus the number it is approaching]). For every number, ε > 0, there is some number, δ > 0; such that, if 0 < x  c < δ; then, f(x)  L < ε. In other words, if the distance between x and c is less than δ; then, the distance between f(x) and L is less than ε.
For example; the limit of (3x  2), as x approaches 3, is 7; f(3) = 7. If one determines that the absolute value [of the function, at x, minus its limit (as x approaches 3)], should be less than 0.003 (that is, one is attempting to determine what value of x will generate an f(x) within 0.003 of the limit of f(x), as x approaches c = 3); then, one can write: (3x  2)  7 = 3x  9 = 3x  3 < 0.003. Noting that 0 < x  c < δ and that, in this situation, c = 3; we can write 0 < x  3 < δ and since 3x  3 < 0.003; it is only logical to conclude that x  3 < 0.003 / 3 = 0.001; and thus, any value within 0.001, of 3, will produce a value within 0.003 of f(x)'s limit (as x approaches 3); that is, a value within 0.003 of 7. For instance, f(3.001)=7.003.
The rest of this article presents the concept of limit in increasing generality, starting with sequences and functions of real numbers, then metric spaces, and culminating in the most general concept, limits of nets on topological spaces. This follows a more and more general definition of how "close to" is evaluated.
Suppose x_{1}, x_{2}, ... is a sequence of real numbers. 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 value x_{n}  L can be interpreted as the "distance" between x_{n} 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.
A function f : R > R is continuous if and only if it is compatible with limits in the following sense:
A subsequence of the sequence (x_{n}) is a sequence of the form (x_{a(n)}) where the a(n) are natural numbers with a(n) < a(n+1) for all n. Intuitively, a subsequence omits some elements of the original sequence. A sequence is convergent if and only if all of its subsequences converge towards the same limit.
The limit operation is a linear operator in the following sense: if (x_{n}) and (y_{n}) are convergent sequences of real numbers and lim x_{n} = L and lim y_{n} = P, then the sequence (x_{n} + y_{n}) is also convergent and has limit L + P. If a is a real number, then the sequence (a x_{n}) is convergent with limit aL. Therefore, the set c of all convergent sequences of real numbers is a real vector space and the limit operation is a linear operator from c to the real numbers.
If (x_{n}) and (y_{n}) are convergent sequences of real numbers with limits L and P respectively, then the sequence (x_{n}y_{n}) is convergent with limit LP. If neither P nor any of the y_{n} is zero, then the sequence (x_{n}/y_{n}) is convergent with limit L/P.
Every convergent sequence is a Cauchy sequence and hence bounded. If (x_{n}) is a bounded sequence of real numbers which is increasing (i.e. x_{n} ≤ x_{n+1} for all n), then it is necessarily convergent. More generally, every Cauchy sequence of real numbers has a limit, or short: the real numbers are complete.
A sequence of real numbers is convergent if and only if its limit inferior and limit superior coincide and are both finite.
Suppose f : U > R is a function, where U is a subset of the real numbers. If p and L are two real numbers, we say that the limit of f(x) as x approaches p is L and write
if and only if
This is equivalent to saying
Note that the function does not have to be defined at p, and in any case, the value f(p) is irrelevant for the determination of the limit of f at p.
We also consider situations where either p or L or both are positive or negative infinity. We say that f(x) approaches positive infinity (+∞) as x approaches p if and only if
We say that the limit of f(x) as x approaches positive infinity is L if and only if
Finally, we say that the limit of f(x) is positive infinity as x approaches positive infinity if and only if
Occasionally, it is useful to approach the point p only from one side. The onesided limit of f(x) as x approaches p from the right is L, written as
if and only if
Leftsided limits are obtained by replacing x  p in the last definition by p  x. By replacing ε by S as above, we can also define onesided limits that are infinite.
If the limit of f(x) as x approaches p exists (which need not be the case), and if there exists at least one sequence (x_{n}) with elements in U  {p} and limit equal to p, then the limit of f(x) as x approaches p is uniquely determined by f and p.
The twosided limit of f(x) as x approaches p exists if and only if the leftsided and rightsided limits exist and are equal.
The function f is continuous at the point p if and only if the twosided limit of f(x) as x approaches p is finite and equal to f(p).
Taking the limit of functions is compatible with the algebraic operations: If
then
and
and
(the latter provided that f_{2}(x) is nonzero in a neighborhood of p and L_{2} is nonzero as well).
These rules are also valid for onesided limits, for the case p = ±∞, and also for infinite limits using the rules
Note that there is no general rule for the case q / 0; it all depends on the way 0 is approached. Some cases, for instance 0/0, 0×∞ ∞∞ or ∞/∞, are also not covered by these rules but the corresponding limits can usually be determined with l'Hôpital's rule.
The real numbers form a metric space if we use the distance function given by the absolute value: d(x,y) = x  y. The same is true for the complex numbers. Furthermore, the Euclidean space R^{n} forms a metric space with the metric given by the euclidean distance. These three will be our motivating examples for extending the limit definitions given above.
If (x_{n}) is a sequence in the metric space (M, d), we say that the sequence has limit L iff for every ε>0 there exists a natural number n_{0} such that for all n>n_{0} we have d(x_{n}, L) < ε.
If the metric space (M, d) is complete (which is true for the real and complex numbers and Euclidean space, and all other Banach spaces), then one can establish the convergence of a sequence in M by showing that it is a Cauchy sequence. The advantage of this approach is that one need not know the limit in advance in order to do this.
If M is a real or complex normed vector space, then the limit operation is linear, as explained above for the case of sequences of real numbers.
Now suppose f : M > N is a map between two metric spaces, p is an element of M and L is an element of N. We say that the limit of f(x) as x approaches p is q and write
if and only if
This is equivalent to saying
The function f is continuous at p if and only if the limit of f(x) as x approaches p exists and is equal to f(p). Equivalently, f transforms every sequence in M which converges towards p into a sequence in N which converges towards f(p).
Again, if N is a normed vector space, then the limit operation is linear in the following sense: if the limit of f(x) as x approaches p is L and the limit of g(x) as x approaches p is P, then the limit of f(x) + g(x) as x approaches p is L + P. If a is a scalar from the base field, then the the limit of af(x) as x approaches p is aL.
If N is R, we can define infinite limits; if M is R, we can define onesided limits in analogy to the definitions given earlier.
All of the above notions of limit can be unified and generalized to arbitrary topological spaces by introducing nets and defining their limits. The article on nets elaborates on this.
An alternative is the concept of limit for filters on topological spaces.
Search Encyclopedia

Featured Article
