y = f (x)
x
y
L
L +
L a
a +a MATH
163, Spring 2001 Due Date: Name(s):
Project 2.4a: The Precise Definition of a Limit
Objective
To investigate the precise definition of limit.
Narrative
If you have not already done so, read Section 2.4 of the text.
To prove that the number L which we guess to be the limit
of f(x) at x = a really is the limit of f(x) at x = a, we must
verify the condition in the formal definition of limit. This
condition requires that for each real number > 0, there is a
real number  > 0 such that the values of f(x) for all x in the
interval (a - , a + ) -- except possibly at x = a itself -- lie
between L - and L + . In this project we investigate the
graphical implications of this condition.
In this project we also introduce the command with(<package name>) which loads the package of routines
named <package name> into Maple. We also illustrate how to draw a line segment from the point P(a, b) to
the point Q(c, d) using the command <segment name> := [[a,b],[c,d]], and the use of plotting options
such as color=blue and scaling=constrained. Finally, we illustrate "delayed plotting" and the use of the
plots command display.
Task
a) Type the command lines below into Maple in the order in which they are listed. These commands are
concerned with limx2(-x3
/12 + x2
/2 + 5/3). Note how we terminate the plot0 := ... and plot1 :=
... lines with a colon ":" rather than a semicolon ";": this suppresses the immediate display of these plot
structures; later we can display them using the plots command display.
> # Project 2.4a: The Precise Definition of a Limit
> restart;
> with(plots):
> f := x -> -x^3/12+x^2/2+5/3;
> a := 2.0;
> L := limit(f(x),x=a);
> xeqa := [[a,0],[a,4]];
> plot0 := plot({f(x),L,xeqa},x=-1..4,y=0..4,color=blue,scaling=constrained):
> display(plot0);
> e := 0.5;
> plot1 := plot({L-e,L+e},x=-1..4,y=0..4,color=red,scaling=constrained):
> display({plot0,plot1});
b) Continue by typing the command lines below into Maple in the order in which they are listed.
> e := 0.2;
> plot1 := plot({L-e,L+e},x=-1..4,y=0..4,color=red,scaling=constrained):
> display({plot0,plot1});
At this point, make a hard-copy of your typed input and Maple's responses. Then, ...
c) Label the graphs of y = f(x), y = L, y = L  e, and x = a on the second graphic you produced in part
(a) by hand. Estimate by eye and state a value of d for which the values of f(x) for all x in the interval
a-d..a+d -- except possibly at x = a -- lie between L-e and L+e when e = 0.5. Draw the lines whose
equations are x = a+d and x = a-d by hand on the second graphic you drew in part (a).
d) Label the graphs of y = f(x), y = L, y = L  e, and x = a on the graphic you produced in part (b) by
hand. Estimate by eye and state a value of d for which the values of f(x) for all x in the interval a-d..a+d
-- except possibly at x = a -- lie between L-e and L+e when e = 0.2. Draw the lines whose equations are
x = a+d and x = a-d by hand on this graphic.
Comments
1. In this project we are not actually proving that L = limxa f(x). On one hand, we are just verifying
that an appropriate d exists for two given e's: to verify that L = limxa f(x), we would have to do
this for every e, not just two, three, four, or any finite number of e's. On the other hand, since Maple
draws the graphs of functions by "connecting-the-dots", some significant behavior could occur between
the dots that is not revealed by Maple, so we cannot trust Maple's graphics to be completely accurate.
This is one of the big reasons the -analysis of limits is so important.
2. You have to tell Maple you want to use a package of routines, such as plots, by saying with(plots)
only once at the beginning (or right after the restart) of a session -- not every time you use a routine
in the package.
3. If the same options are to be used several times in a Maple session, they can be specified once at
the beginning of the session using the setoptions command and omitted thereafter (saving time and
reducing typing). For example, if we had included the command
> setoptions(color=blue,scaling=constrained):
immediately after the with(plots) command, then in all subsequent plots we would have automatically
declared color=blue and scaling=constrained.