Estimation and Limits -- Dividing Functions

You should use one of the computer algebra systems below with this module. Click on the appropriate icon for your preferred CAS and then arrange your screen so that you can easily move back-and-forth between this window and your CAS window. Click on the appropriate help button for help.

This is the fourth in a series of modules that deal with the ways in which functions can be changed or combined. In this module we look at what happens when one function is divided by another function.

This is useful in many different contexts. For example, if the function f(t) describes the total food supply in a particular country at time t and the function p(t) describes the population of the country then the quotient C(t) = f(t)/p(t) describes the per capita food supply.

Use your CAS window to compare the following functions

f(t) = 1000 + 50 t
p(t) = 1000 * 1.02^t
h(t) = f(t) / p(t)

Use your CAS window to compare the following functions

f(t) = sin 2 pi t
g(t) = (t^2 + 1)
h(t) = f(t) / g(t)

The basic fact that we discuss here is that if

  Lim  f(x) = L
x --> a

and

  Lim  g(x) = M
x --> a

then unless M = 0

  Lim  f(x) / g(x) = L / M
x --> a

We can express this same fact by saying that if f(x) is close to L when x is close to a and if g(x) is close to M when x is close to a then (unless M = 0 ) f(x) / g(x) is close to L / M when x is close to a.

However we express it, this descriptive fact is only part of a more interesting story. The first hypothesis -- the fact that

  Lim  f(x) = L
x --> a

requires that we can make f(x) as close as we want to L by making x sufficiently close to a. The second hypothesis -- the fact that

  Lim  g(x) = M
x --> a

requires that we can make g(x) as close as we want to M by making x sufficiently close to a. The conclusion -- the fact that

  Lim  f(x) / g(x) = L / M
x --> a

tells us that we can make f(x) / g(x) as close as we want to L / M by making x sufficiently close to a.

Check Your Understanding

Show that for any function r(x) = p(x) / q(x) where p(x) and q(x) are polynomials we have (unless q(a) = 0).
```
  Lim   r(x) = r(a)
x --> a
```
Hint: See the Check Your Understanding section of the module on Multiplying Functions.
Functions like r(x) are called rational functions.
answer
Explain why this modules has all the parenthetical restrictions (unless ___ = 0).
answer
A manufacturer wants to make a connecting rod whose length is 100 cm with a tolerance of 0.1 cm. The rod will be made by pouring molten metal into a cylindrical mold. The process has two important components -- the mold and a pouring device that measures the molten metal poured into the mold. The perfect mold would have a cross section of 1 square centimeter and the perfect pouring device would pour exactly 100 cubic centimeters of molten metal. The perfect rod would have a length of exactly 100 centimeters. The manufacturer has the following choices of molds and pouring devices.
```
---------------------
         Molds

 Tolerance      Cost

0.010  cm^2    $10.00
0.005  cm^2    $25.00
0.002  cm^2    $40.00
0.001  cm^2    $85.00
---------------------

     Pouring Devices

Tolerance     Cost

1.0  cm^3     $15.00
0.5  cm^3     $30.00
0.2  cm^3     $60.00
0.1  cm^3     $95.00
----------------------
```
What should the manufacturer do?
answer