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单词 ImproperLimits
释义

improper limits


In calculus there is often used such expressions as “the limit of a functionMathworldPlanetmath is infinite”, and one may write for instance that

limx01x2=.

Such “limits” are actually of the limit notion, and can be defined exactly.  They are called improper limits.

Definition.  Let the real function f be defined in a neighbourhood of the point x0.

limxx0f(x)=

iff for every real number M there exists a number δM such that

f(x)>M

as soon as

0<|x-x0|<δM.

In a similar way we can define the improper limit - of a real function.  The definition may be extended also to the cases  x±, when one speaks of limits at infinity.

Note 1.  If  limxx0f(x)=  and limxx0g(x)=a>0,  then we have

limxx0f(x)g(x)=.

Hence we can say that  a=  when  a>0.  There are some other “mnemonics of infinite” (cf. the extended real numbers):

a=-  (a< 0)
±+a=±
a±= 0
+=
=
-=-

On the contrary, there exist no mnemonics for the cases

0,-,,00,  00,0,  1;

they are and depend on the instance (cf. the indeterminate form).

Note 2.  In the complex planeMathworldPlanetmath, the expression

limzz0f(z)=

means that  limzz0|f(z)|=.

Titleimproper limits
Canonical nameImproperLimits
Date of creation2013-03-22 14:40:45
Last modified on2013-03-22 14:40:45
Ownerpahio (2872)
Last modified bypahio (2872)
Numerical id24
Authorpahio (2872)
Entry typeDefinition
Classificationmsc 26A06
Synonyminfinite limits
Synonymimproper limit
Related topicLHpitalsRule
Related topicExtendedRealNumbers
Related topicLimitRulesOfFunctions
Related topicIntegratingTanXOver0fracpi2
Related topicIndeterminateForm
Related topicExampleOfJumpDiscontinuity
Related topicListOfCommonLimits
Related topicLimitsOfNaturalLogarithm
Related topicSecondDerivativeAsSimpleLimit
Related topicAngleBetweenTwoLines
Defineslimit at infinity
Definesmnemonic of infinite
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更新时间:2025/5/4 19:54:18