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


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.


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


as soon as


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


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±= 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


means that  limzz0|f(z)|=.

Title improper limits
Canonical name ImproperLimits
Date of creation 2013-03-22 14:40:45
Last modified on 2013-03-22 14:40:45
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 24
Author pahio (2872)
Entry type Definition
Classification msc 26A06
Synonym infinite limits
Synonym improper limit
Related topic LHpitalsRule
Related topic ExtendedRealNumbers
Related topic LimitRulesOfFunctions
Related topic IntegratingTanXOver0fracpi2
Related topic IndeterminateForm
Related topic ExampleOfJumpDiscontinuity
Related topic ListOfCommonLimits
Related topic LimitsOfNaturalLogarithm
Related topic SecondDerivativeAsSimpleLimit
Related topic AngleBetweenTwoLines
Defines limit at infinity
Defines mnemonic of infinite