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Joe Joyner's Homepage - Norview High School

Joe Joyner
Mathematics Teacher
Norview High School
Norfolk, VA
(757) 852-4500
jjoyner@nps.k12.va.us

Limit of A Function

    In the study of calculus, understanding the concept of the limit of a function is central to development of the concept of a derivative of a function.  You may approach the concept of the limit of a function from one of three perspective - numerically, graphically and algebraically.

    In the numerical approach , the use of a graphing calculator, mathematics software, spreadsheet or similar utility will prove helpful.  Of course, one may work the calculations with the aid of only a scientific calculator.  Given a function,  f(x) , to observe the values that  f(x) assumes as  x  approaches a value,  c , calculate the values of  f(x) as   gets closer and closer in value to  c.  This "inching up" to the value   must be done with values of  x approaching   from below (that is, increasing values of  x that are less than  c ) as well as with values of   approaching   from above (that is, decreasing values of  x  that are greater than  c ).  A table of values can be made and examined to determine whether the values of   f(x) appear to be approaching the same value as  x  approaches   from above and below.  This method though only allows for an estimation of the limit for all but the simplest cases of  f(x).

    To see a tutorial on estimating limits numerically, go to Limits and Continuity: Numerical Approach .


    In the graphical approach , the use of a graphing calculator or mathematics software that allows the user to "trace" along the function's graph will prove helpful.  One may graph the function manually, but this is more time consuming, especially for more complex functions.  Given a function,  f(x) , to observe the values that  f(x)  assumes as  x  approaches a value, c , graph the function and trace along it as the x-coordinates approach the value  c .  This "inching up" to the value   must also be done with values of  x  approaching  c  from the left (that is, increasing values of  x  that are less than  c ) as well as with values of  x approaching   from the right (that is, decreasing values of   x  that are greater than  c ).  This method also only allows for an estimation of the limit of most functions.

    For an excellent treatment of estimating limits graphically, go to  Limits and Continuity: Graphical Approach .


    In the algebraic approach , formal algebraic (including arithmetic) manipulation of  f(x) is used to find an exact representation of the the limit of  f(x) .   To see a tutorial on calculating limits algebraically, go to Limits and Continuity: Algebraic Approach .
 

    If you'd care to discuss this or similar topics, email me at Norview High School .
  


Updated Dec 23, 2002 
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