limit rules of functions

Theorem 1.

Let f and g be two real ( or complex functions.  Suppose that there exist the limits  limxx0f(x)  and  limxx0g(x).  Then there exist the limits  limxx0[f(x)±g(x)],  limxx0f(x)g(x)  and, if  limxx0g(x)0,  also  limxx0f(x)/g(x), and

  1. 1.


  2. 2.


  3. 3.


  4. 4.


These rules are used in limit calculations and in proving the corresponding differentiation rules (sum ruleMathworldPlanetmath, product ruleMathworldPlanetmath etc.).

In 1, the domains of f and g could be any topological spaceMathworldPlanetmath (not necessarily or ).

There are limit rules of sequences (

As well, one often needs the

Theorem 2.

If there exists the limit  limxx0f(x)=a  and if g is continuous at the point  x=a, then there exists the limit  limxx0g(f(x)), and

Title limit rules of functions
Canonical name LimitRulesOfFunctions
Date of creation 2013-03-22 14:51:21
Last modified on 2013-03-22 14:51:21
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 20
Author pahio (2872)
Entry type Theorem
Classification msc 26A06
Classification msc 30A99
Synonym limit rules of sequences
Related topic GrowthOfExponentialFunction
Related topic ImproperLimits
Related topic DerivativesOfSineAndCosine
Related topic ListOfCommonLimits
Related topic LimitExamples
Related topic ProductAndQuotientOfFunctionsSum
Related topic DerivationOfPlasticNumber