Mathematical limit

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The mathematical concept of limit is used to describe the behaviour of a function as its arguments get close to a given point, or the behavior of the elements of a sequence as the index approaches infinity. Limits are very important in calculus and analysis. They serve to define the concepts of derivative and continuity.


Limit of a sequence

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

for every ε>0 there exists a natural number N (which will depend on ε) such that for all n>N we have d(xn, x) < ε.

Intuitively, this means that eventually, all elements of the sequence get as close as we want to the limit. 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.

Examples:

  • The sequence 1/1, 1/2, 1/3, 1/4, ... of real numbers converges with limit 0.
  • If z is a complex number with |z| < 1, then the sequence z, z2, z3, ... of complex numbers converges with limit 0. Geometrically, these numbers "spiral into" the origin.
  • The sequence 1, -1, 1, -1, 1, ... is divergent.
  • The sequence 1/2, 1/2 + 1/4, 1/2 + 1/4 + 1/8, 1/2 + 1/4 + 1/8 + 1/16, ... converges with limit 1. This is an example of an infinite series.
  • In the metric space C[a,b] of all continuous functions defined on the interval [a,b], with distance arising from the supremum norm, every element can be written as the limit of a sequence of polynomial functions. This is the content of the Stone-Weierstrass theorem.

Properties

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 xnxn+1 for all n, then it is necessarily convergent.

Limit of a function at a point

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

for every ε > 0 there exists a δ > 0 such that whenever 0 < |x - p| < δ then |f(x) - q| < ε.

This is equivalent to saying

for every convergent sequence (xn) with elements in U - {p} and limit equal to p, the sequence (f(xn)) converges with limit q.

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

for every M > 0 there exists a δ > 0 such that whenever 0 < |x - p| < δ then f(x) > M.

We say that the limit of f(x) as x approaches positive infinity is q if and only if

for every ε > 0 there exists an N > 0 such that whenever x > N, then |f(x) - q| < ε.

Finally, we say that the limit of f(x) is positive infinity as x approaches positive infinity if and only if

for every M > 0 there exists an N > 0 such that whenever x > M, then f(x) > N.

The definitions for negative infinity are analogous.

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.

Examples:

  • The limit of 1/x as x approaches infinity is 0.
  • The limit of x2 as x approaches 3 of is 9. (In this case the value of the function happens to be well defined at the point, and the value is the same as the limit.)
  • The limit of xx as x approaches 0 is 1.
  • The limit of ((a + x)2 - a2 ) / x as x approaches 0 is 2a.
  • The limit of x sin(1/x) as x approaches positive infinity is 1.
  • The limit of (cos(x) - 1)/x as x approaches 0 is 0.

Properties

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→p

then

   lim  (f1(x) + f2(x))  =  q1 + q2
   x→p

and

   lim  (f1(x) * f2(x))  =  q1 * q2
   x→p

and

   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

  • q + ∞ = ∞ for q ≠ -∞
  • q * ∞ = ∞ if q>0
  • q * ∞ = -∞ if q<0
  • q / ∞ = 0 if q ≠ ± ∞

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.

Generalizations

Mention limits of filters and nets in arbitrary topological spaces