Here, we will explain the concept of limit for metric spaces, always keeping in mind that the real and complex numbers form a metric space with the distance function given by the absolute value: d(x,y) = |x - y|. Furthermore, the Euclidean space Rn forms a metric space with the metric given by the euclidean distance. These three will be our motivating examples.
Suppose x1, x2, ... is a sequence of elements of a metric space (M, d). We say that x is the limit of this sequence and we write
lim xn = x n → ∞if and only if
Every convergent sequence is bounded.
A subsequence of the sequence (xn) is a sequence of the form (xa(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.
Taking as our metric space a normed vector space V (the real numbers, complex numbers and Euclidean space are examples), then the limit operation becomes a linear operator: if (xn) and (yn) are convergent sequences with elements in V and lim xn = x and lim yn = y, then the sequence (xn + yn) is also convergent with limit x + y. If a is a scalar from the base field, then the sequence (a xn) is convergent with limit ax. Therefore, the set c of all convergent sequences in V is a vector space and the limit is a linear operator from c to the base field.
If (xn) and (yn) are convergent sequences of real or complex numbers with limits x and y respectively, then the sequence (xnyn) is convergent with limit xy. If neither y nor any of the yn is zero, then the sequence (xn/yn) is convergent with limit x/y.
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 (xn) is a bounded sequence of real numbers such that xn ≤ xn+1 for all n, then it is necessarily convergent.
Suppose U and V are subsets of the real or complex numbers, f : U -> V is a function, and p is a real or complex number. We say that the limit of f(x) as x approaches p is q and write
lim f(x) = q x→p
if and only if
This is equivalent to saying
Note that the function f does not have to be defined at the point p and in any event, the function value f(p) is irrelevant for the determination of its limit at p.
For a real function f we allow both p and q to be 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 q 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
In all of the above cases, one can show that if the limit exists (which need not be the case), and if there exists at least one sequence (xn) with elements in U - {p} and limit equal to p, then the limit is uniquely determined by f and p.
The function f is continuous at the point p if and only if the limit of f(x) as x approaches p is f(p).
Taking the limit of functions is compatible with the algebraic operations: If
lim f1(x) = q1 and lim f2(x) = q2 x→p x→pthen
lim (f1(x) + f2(x)) = q1 + q2 x→pand
lim (f1(x) * f2(x)) = q1 * q2 x→pand
lim (f1(x) / f2(x)) = q1 / q2 x→p(the latter provided that f2(x) is non-zero in a neighborhood of p and q2 is non-zero as well).
These rules are also valid for infinite limits, using the rules
Some cases, for instance 0/0, 0*∞ or ∞/∞, are not covered by these rules; the corresponding limits can usually be determined with l'Hôpital's rule.