limit examples

Example 1.  Determine the limit limx0tanxx.  — Using the definition of tan and the limit rule of product ( we can write


The limit in the former factor ( is 1 by the parent entry ( 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


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


which is the circumference of the circle.

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


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


Example 5.  One may use the definition of derivativePlanetmathPlanetmath in

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