list of common limits

Following is a list of common limits used in elementary calculus:

  • For any real numbers a and c,  limxac=c.

  • For any real numbers a and n,  limxaxn=an  (proven here ( for n a positive integer)

  • limx0sinxx=1  (proven here (

  • limx01-cosxx=0  (proven here (

  • limx0arcsinxx=1  (proven here (

  • limx0ex-1x=1  (proven here (

  • For a>0,  limx0ax-1x=lna (proven here (

  • For b>1 and a any real number,  limxxabx=0  (proven here (

  • limx0+xx=1  (proven here (

  • limx0+xlnx=0  (proven here (

  • limxlnxx=0  (proven here (

  • limxx1x=1  (proven here (

  • limx±(1+1x)x=e

  • limx0(1+x)1x=e

  • limx0(1+sinx)1x=e  (power of e, l’Hôpital’s rule (

  • limx(x-x2-a2)=0  (proven here (

  • For a>0 and n a positive integer,  limxax-axn-an=1nan-1.

  • limx0tanx-sinxx3=12  (by l’Hôpital’s rule (

  • For q>0, limx(logx)pxq=0

  • tan(x+π2)=limξπ2tanx+tanξ1-tanxtanξ=limξπ2sec2ξ-tanxsec2ξ=-cotx    (by  l’Hôpital’s rule (
    That is, tanxtan(x+π2)=-1, which indicates orthogonality of the slopes represented by those functions.

  • For a real or complex constant c and a variable z,

  • For x real (or complex),  limnn(xn-1)=logx  (proven here ( for real x).



  • 1 Catherine Roberts & Ray McLenaghan, “ContinuousMathworldPlanetmathPlanetmath Mathematics” in Standard Mathematical Tables and Formulae ed. Daniel Zwillinger. Boca Raton: CRC Press (1996): 333, 5.1 Differential CalculusMathworldPlanetmath
Title list of common limits
Canonical name ListOfCommonLimits
Date of creation 2014-02-23 10:09:07
Last modified on 2014-02-23 10:09:07
Owner Wkbj79 (1863)
Last modified by pahio (2872)
Numerical id 29
Author Wkbj79 (2872)
Entry type Feature
Classification msc 26A06
Classification msc 26A03
Classification msc 26-00
Related topic LimitRulesOfFunctions
Related topic ImproperLimits
Related topic LimitExamples
Related topic HalleysFormula