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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 x gets closer and closer in value
to c. This "inching up" to the value
c must be done with values of x approaching
c from below (that is, increasing values of x that
are less than c ) as well as with values of x
approaching c 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
c 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 c 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
c 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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