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Introduction to Limits

limits1

The word limit in this topic represents the limiting value of a function at a given point.  In order to learn continuity and differentiability, we must be familiar with the topic ‘limits’.  Here we learn some standard results in limits, algebra of limits, theorems related to limits etc.

Consider the function f(x) = 3x2.  We can see that as ‘x’ takes the value very close to 0, the value of f(x) also moves towards 0, so we say
         limits2 
          
We will read this as limit x tends to zero f(x) equals zero 
In general, as limits3 then ‘l’ is called limit of the function f(x) which is symbolically written as  limits4
In some cases the function f(x) takes different values to the left of ‘a’ and right of ‘a’ so we have to find the left hand limit as well as right hand limit.
We say limits5 is the expected value of ‘f’ at x=a given the values of f near
                                             
‘x’ to the left of a.  This value is called the left hand limit of f at ‘a’   
Similarly limits5 is the expected valu of ‘f’ at x=a given the values of f near
           
‘x’ to the right of ‘a’.  This value is called the right hand limits of ‘f’ at ‘a’.
If the right and left hand limits coincide, we call that common value as the limit of f(x) at x=a and denote it by limits6.
                                                  

Example: limits7

Algebra of limits


Let ‘f’ and ‘g’ be two functions such that both limits8 exist.
                                                                
then the following are algebra of limits.
1)    Limit of sum of two functions is the sum of the limits of the functions.
            limits9
2)    Limit of difference of two functions is difference of the limits of the functions.
              limits10


  1. 3)    Limit of the product of two functions is product of the limits of the functions.
                  limits11
    4)    Limit of quotient of two functions is quotient of the limits of the function [provided denominator is non-zero].
                   limits12

    Limits of polynomials and rational functions


    A function f is said to be a polynomial function if f(x) is zero function of if f(x) =a0+a1x+a2x2+………+anxn, where ai’s are real numbers such that an≠0 for some natural number n.
    We have, limits13
    A function is said to be a rational function, if f(x) = g(x), where g(x) and h(x) are polynomials such that h(x)≠0                                   ____
  2.                                                                           h(x)
    limits14

    Limits of Trigonometric Functions


    The following stated theorems about functions in general come in handy in calculating limits of some trigonometric functions.
    •    Let f and g be two real valued functions with the same domain such that  f(x) ≤ g(x) for all x in the domain of definitions, for some ‘a’, if both          
  3.         limits15
    •    Sandwich Theorem:  Let f, g and h be real functions such that f(x)≤g(x)≤h(x) for all x in the common domain of definition.  For some real number ‘a’ , if
          limits16
    The geometric proof of an important inequality relating trigonometric functions is given below.
                    limits17
    We know that sin(-x) = -sinx and cos(-x) = cosx.  Hence, it is sufficient to prove the inequality for 0<x<π/2
    “O” is the center of the unit circle such that angle AOC is ‘x’ radians and 0<x<π/2.  Line segments BA and CD are perpendiculars to OA.  Join AC. Then,                                                                                               
    limits18Area of ∆OAC < Area of sector OAC < Area of ∆OAB                          
    ½ OA.CD  <  (x/2π). Π(OA)2 < ½ OA.OB                                             
    CD  <  x . OA < AB                                                            
    From ∆OCD,
    Sinx = CD/OA (since CD=OA) and hence CD=OA Sinx.  Also tanx = AB/OA and therefore, AB = OA. Tanx.  Thus,
    OA.  sinx < OA. X < OA. tanx
    Since OA is the length, it is positive, we have
    sinx < x < tanx
    Since 0<x<π/2, sinx is positive and dividing throughout by sinx, we have
    1 < x/sinx < 1/cosx
    Taking reciprocals, cosx < sinx < 1
                                       ____                               
  4.                                      x

    Some Standard Results in Limits


    limits19
                                      
    Now try it yourself!  Should you still need any help, click here to schedule live online session with e Tutor!

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