View source for Double limit

The double limit of a sequence, the limit of a double sequence, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d0338901.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d0338902.png" /> is a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d0338903.png" /> which is defined as follows: For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d0338904.png" /> there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d0338905.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d0338906.png" /> the equality

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is valid. The notation is

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If for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d0338909.png" /> there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389010.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389011.png" /> the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389012.png" /> is fulfilled, then the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389013.png" /> has infinity as its limit:

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The infinite limits

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are defined in the same manner. The double limit of a sequence is a special case of the double limit of a function over a set, namely when this set consists of the points on the plane with integer coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389017.png" />. Accordingly, the double limit of a sequence is connected with its repeated limits as in the general case.

The double limit of a function is the limit of a function of two variables, defined as follows. Let the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389018.png" /> be defined on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389019.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389020.png" />-plane, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389021.png" /> be a limit point of it (cf. [[Limit point of a set|Limit point of a set]]). A number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389022.png" /> is said to be the double limit of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389023.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389024.png" />, or as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389025.png" />, if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389026.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389027.png" /> such that for all the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389028.png" /> the coordinates of which satisfy the inequalities

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the inequality

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is valid. In such a case the limit is written as

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The double limit of a function may be formulated in terms of the limit of a sequence:

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if for any sequence

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the condition

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is satisfied. The double limit of a function as its arguments tend to infinity, as well as the definitions of infinite double limits of a function, are formulated in a similar manner. There exists a connection between the double limit of a function and the [[Repeated limit|repeated limit]] of a function at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389036.png" /> or at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389037.png" />: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389039.png" /> be limit points (finite or infinite) of the real subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389041.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389042.png" />. If for a function a finite or infinite double limit

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exists, and if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389044.png" /> there is a finite limit

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then the repeated limit

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exists and is equal to the double limit of the function.

Using the concept of a neighbourhood, the following form may be assigned to the definition of the double limit of a function: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389047.png" /> be a limit point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389048.png" /> of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389049.png" /> or the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389050.png" />, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389051.png" /> being unbounded in the latter case, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389052.png" /> be a number or one of the symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389055.png" />. Then

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if for any neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389057.png" /> of the point or symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389058.png" /> there exists a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389059.png" /> of the number or symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389060.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389062.png" />, the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389063.png" /> is satisfied. In this form the definition of the double limit of a function is applied to the case when the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389064.png" /> is defined on the product of two topological spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389065.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389066.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389067.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389068.png" />, while the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389069.png" /> also belong to a topological space.