Talk:Limit of a function

To be merged with the main article:

Metric spaces

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 Rⁿ 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 n₀ such that for all n>n₀ 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

<math> \lim_{x \to p}f(x) = q <math>

if and only if to be merged into the main article:

for every ε > 0 there exists a δ > 0 such that for all x in M with 0 < d(x, p) < δ, we have d(f(x), L) < ε.

This is equivalent to saying

for every convergent sequence (x_n) in M - {p} with limit equal to p, the sequence (f(x_n)) converges with limit L.

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 one-sided limits in analogy to the definitions given earlier.

Do multiple limits commute?

Hi,

One piece of information I was looking for but couldn't find on wikipedia was whether multiple limits commute. That is, given a function <math>f(x,y)<math>, is it necessarily the case that

<math>\lim_{x \to x_0} \lim_{y \to y_0} f(x,y) = \lim_{y \to y_0} \lim_{x \to x_0} f(x,y)<math>

? I think this could be added to the article, but I couldn't find it in any of the places I checked online. Thanks! -- Creidieki 05:57, 20 Sep 2004 (UTC)

Limit don't necessary commute. For instance,

<math>\lim_{y\to0} \frac{x}{x^2+y^2} = \frac1x \mbox{, so } \lim_{x\to0} \lim_{y\to0} \frac{x}{x^2+y^2} \mbox{ diverges;} <math>

<math>\lim_{x\to0} \frac{x}{x^2+y^2} = 0 \mbox{ for } y \neq 0 \mbox{, so } \lim_{y\to0} \lim_{x\to0} \frac{x}{x^2+y^2} = 0. <math>

I believe there is a theorem that the limits do commute in some cases, perhaps somebody else remembers the details. And yes, this would make a good addition to the article. -- Jitse Niesen 15:37, 20 Sep 2004 (UTC)