limit examples


Example 1.  Determine the limit limx0tanxx.  — Using the definition of tan and the limit rule of product (http://planetmath.org/LimitRulesOfFunctions) we can write

limx0tanxx=limx0(sinxx1cosx)=limx0sinxxlimx01cosx.

The limit in the former factor (http://planetmath.org/ProductPlanetmathPlanetmathPlanetmath) is 1 by the parent entry (http://planetmath.org/LimitOfDisplaystyleFracsinXxAsXApproaches0). Also the latter limit is 1, since  cosx and thus the quotient 1cosx is continuousMathworldPlanetmathPlanetmath in the point  x=0 (see continuity of sine and cosine).  Accordingly the desired limit is 1.

Example 2.  Determine the limit limx0sinaxsinbx.  — As above, we can write

limx0sinaxsinbx=limx0(sinaxaxbxsinbxab)=limx0sinaxaxlimx0bxsinbxlimx0ab=11ab=ab.

Example 3.  The perimeterPlanetmathPlanetmath of a regularPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath n-gon, circumscribedMathworldPlanetmath to a circle with radius 1, is 2ntanπn.  Determine the limit of this perimeter as n tends to infinityMathworldPlanetmath.  — Utilising the example 1 we can calculate

limn2ntanπn=limn2πtanπnπn=2π1=2π,

which is the circumference of the circle.

Example 4.  Determine the limit limx0arcsinxx.  — If we denote

arcsinx:=y,

the monotonicity of the arcus sine (http://planetmath.org/CyclometricFunctions) function on  [-1, 1]  implies that “x0” is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath (http://planetmath.org/Equivalent3) to “y0”.  Then we can calculate:

limx0arcsinxx=limy0ysiny=limy01sinyy=11=1.

Example 5.  One may use the definition of derivativePlanetmathPlanetmath in

limx0arctanxx=limx0arctanx-arctan0x-0=[ddxarctanx]x=0=11+02=1.
Title limit examples
Canonical name LimitExamples
Date of creation 2013-03-22 17:40:16
Last modified on 2013-03-22 17:40:16
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Example
Classification msc 26A06
Classification msc 26A03
Synonym utilizing limit of sinxx in 0
Related topic LimitRulesOfFunctions
Related topic DerivativeOfInverseFunction
Related topic ListOfCommonLimits