/* [wxMaxima batch file version 1] [ DO NOT EDIT BY HAND! ]*/
/* [ Created with wxMaxima version 11.08.0 ] */

/* [wxMaxima: title   start ]
Take It to the Limit
   [wxMaxima: title   end   ] */

/* [wxMaxima: comment start ]
Chapter 2 Technology Application Project
   [wxMaxima: comment end   ] */

/* [wxMaxima: comment start ]
In executing this module, you can expect to see an error in the 
evaluation. This will happen whenever you try to divide by 0, as we 
do in certain evaluations.
   [wxMaxima: comment end   ] */

/* [wxMaxima: section start ]
Introduction
   [wxMaxima: section end   ] */

/* [wxMaxima: comment start ]
OBJECTIVE: To interpret the limit concept graphically and 
numerically. 

In this module, you will explore some limits by graphing the 
functions involved and by creating tables of values for the 
functions. You will explore, in some detail, functions that have 
indeterminate forms at the limit point, and you will come to 
appreciate more fully the importance of formal proofs in 
mathematics.
   [wxMaxima: comment end   ] */

/* [wxMaxima: subsect start ]
Technology Guidelines
   [wxMaxima: subsect end   ] */

/* [wxMaxima: comment start ]
NOTE:  If you have just finished a document, restart Maxima 
before executing a new document.  This can be done by choosing
"Restart" from the Maxima menu.

TO OPEN OR CLOSE CELLS
Click on the arrow at the top of the cell bracket. 

TO STOP AN EXECUTION
Click on STOP button from the toolbar. 

ORDER OF EXECUTION
Execute commands in the order given. Do not skip any Maxima
Input lines within a given document. 

Alternatively, you can execute the entire worksheet by 
selecting the "Evaluate All Cells" command from the "Cell" drop
down menu or simply press Ctrl-r. 

SAVING WORKSHEETS
You can save anytime to any directory you choose, and it is 
wise to save often.

EXPERIENCING MAJOR PROBLEMS
Save if appropriate, and then shut down Maxima and start it up 
again. 
   [wxMaxima: comment end   ] */

/* [wxMaxima: section start ]
PART I:  What is 0/0?
   [wxMaxima: section end   ] */

/* [wxMaxima: comment start ]
Section 2.3, Exercise 64
   [wxMaxima: comment end   ] */

/* [wxMaxima: subsect start ]
Observing the Data
   [wxMaxima: subsect end   ] */

/* [wxMaxima: comment start ]
When we try to evaluate a function like (x(1-cos(x))/(x-sin(x)) at 
x=0, we get 0/0,but what does this mean? The answer to the question 
"What is 0/0?" depends on whether the numerator or the denominator 
of the rational function we are trying to evaluate approaches 0 
faster. This function is not defined at x=0. because its denominator
is 0 there; however, the numerator is also 0 at x=0. To better 
understand the situation, let's investigate what happens to the 
values of (x(1-cos(x))/(x-sin(x)) when x is very near 0. We begin 
by graphing the function.
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
load(draw)$
numer:true$
ratprint:false$
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
f (x) := x*(1-cos(x))/(x-sin(x));
wxdraw2d(
    nticks=200,
    color=red,
    explicit(f(x),x,-20,20),
    xlabel="x",
    ylabel="y");
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
The graph seems to indicate that this function is approaching 3 as 
x approaches 0. To see that this is the case, let's compute some 
values of the function for x near 0. The makelist command provides a 
way to do this, and we use it to create a list of entries, 
{x, f(x), f(x)-3} for values ranging from -1 to 1 in increments of 
0.1. Don't be surprised by the error messages that are produced 
when you execute the next command; they are to remind you that f(x) 
is not defined at x=0.
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
count: 1$
for i: -1.0 thru 0 step 0.1 do block(
    mm[count]: [i,f(i),f(i)-3],
    count: count+1)$
matrix1: matrix(["x", "f(x)", "f(x)-3"])$
for j: 1 thru (count-1) step 1 do block(
        matrix1: addrow(matrix1,mm[j]))$
print(matrix1)$
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
count: 1$
for i: 1.0 thru 0 step -0.1 do block(
    mm[count]: [i,f(i),f(i)-3],
    count: count+1)$
matrix2: matrix(["x", "f(x)", "f(x)-3"])$
for j: 1 thru (count-1) step 1 do block(
        matrix2: addrow(matrix2,mm[j]))$
print(matrix2)$
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
Notice that a command terminates immediately upon reaching a divide 
by 0 error.  Therefore, the matrix row for x=0 did not get added.  
So we modify our commands to handle the case when x=0.
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
count: 1$
for i: -1 thru 0 step 1/10 do block(
    if (abs(i)<0.001) then undefined: true else undefined: false,
    if (undefined) then block(
        mm[count]: [i,"undefined","undefined"])
    else block(
        mm[count]: [i,f(i),f(i)-3]),
    count: count+1)$
matrix1: matrix(["x", "f(x)", "f(x)-3"])$
for j: 1 thru (count-1) step 1 do block(
        matrix1: addrow(matrix1,mm[j]))$
print(matrix1)$
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
count: 1$
for i: 1 thru 0 step -1/10 do block(
    if (abs(i)<0.001) then undefined: true else undefined: false,
    if (undefined) then block(
        mm[count]: [i,"undefined","undefined"])
    else block(
        mm[count]: [i,f(i),f(i)-3]),
    count: count+1)$
matrix2: matrix(["x", "f(x)", "f(x)-3"])$
for j: 1 thru (count-1) step 1 do block(
        matrix2: addrow(matrix2,mm[j]))$
print(matrix2)$
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
How close to 0 must I hold x in order to guarantee that f(x) stays 
within 1/100th of 3? You will explore that in the You Try It 
subsection at the end of this section.
   [wxMaxima: comment end   ] */

/* [wxMaxima: subsect start ]
Continuity Issues
   [wxMaxima: subsect end   ] */

/* [wxMaxima: comment start ]
You might notice from the graph of  f(x) that it appears to be 
continuous. For a function to be continuous, the limit as x 
approaches 0 must exist and this condition seems to be met. However,
continuity requires that the function be defined when x is 0, and 
here we have a problem. If we evaluate f(0), the result is an 
indeterminate form (0/0 is such a form).
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
f(0);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
Is f(x) continuous at x=0? why?

The graph of f(x) is deceiving. Mathematically, there is a gaping 
hole in the graph at the point (0,3) even though we can't see it on 
the Maxima graph of f(x). As a result, we might be falsely led to 
believe that f(x) is continuous at x=0. Now consider the function
g(x) = (x(1-cos(x))/(x-sin(x)) if x is not equal to 0 and g(x) = 3
if x=0. What is g(0)? What is the limit as x approaches 0 of g(x)? 
Is this function continuous at x=0?

The discontinuity in f(x) at x=0 is said to be "removable" since we 
can remove it by forming a new continuous function, g(x), that is 
identical to f(x) when x is not equal to 0 and is equal to 3 when 
x=0.

For further exploration of continuous and discontinuous functions, 
refer to "Continuous and Discontinuous Curves," a JAVA applet 
included in this supplement.
   [wxMaxima: comment end   ] */

/* [wxMaxima: subsect start ]
You Try It: PART I
   [wxMaxima: subsect end   ] */

/* [wxMaxima: comment start ]
TAKE THE EPSILON CHALLENGE

We have a challenge for you. Here it is.

Determine how close x must be to 0 so that the values of f(x) will 
be within a specified distance from 3. More specifically, we 
challenge you with a small positive number, let's call it epsilon, 
and ask you try to determine a value for delta so that when 
-delta < x < delta with x not equal to 0, the values of f(x) will 
satisfy the inequality |f(x)-3| < epsilon.  That is, the distance 
between the value of f(x) and 3 will be less than epsilon. For 
example, if epsilon = .1e-1, the table of values in the cell above 
shows that -.1 < f(x)-3 < .1 is not satisfied when delta=1.  You 
need to find a smaller delta that will keep x closer to 0 so that 
you can beat our epsilon challenge.

To help you with this, we include a command in the next cell that 
generates a list of ordered pairs [x, |f(x)-3|] so that you can 
look at the magnitude of the differences between the values of f(x)
and 3 to see if the delta you picked meets our epsilon challenge. 
In addition, we added to each ordered pair the result of a test to 
see if a is within a distance epsilon of 3. We start with 
epsilon = .1e-1 and a value for delta that is too big. You should 
try different values for delta, re-evaluating the commands in the 
cell each time, until you find a delta that works. After you find a
delta that works for epsilon = .1e-1, try to find delta-values that 
work for epsilon = .1e-3, .1e-6, 10(-10), ....
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
delta: 0.0025$
epsilon: 0.01$
L: 3$
count: 1$
for i: -delta thru delta step delta/10 do block(
    if (is(equal(i,0))) then undefined: true else undefined: false,
    if (undefined) then block(result: false, mm[count]: [i,"undefined",result])
    else block(
        if (abs(f(i)-L)<epsilon) then result: true 
            else result: false,
        mm[count]: [i,abs(f(i)-L),result]),
    count: count+1)$
matrix3: matrix(["x", "|f(x)-L|", "result"])$
for j: 1 thru (count-1) step 1 do block(
        matrix3: addrow(matrix3,mm[j]))$
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
print(matrix3)$
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
The formal definition of the limit says that if you are able meet 
our epsilon challenge, no matter how small we make our epsilon and 
if you are able to prove that you can do it, then you can say 
unequivocally that the limit as x approaches 0 of f(x) is 3. We 
aren't going to ask you to prove this limit, but your instructor 
may ask you to prove some simpler ones. One thing you can do, 
however, is use Maxima's limit( ) command to see if it verifies our 
conjecture that the limit as x approaches 0 of f(x) is 3.
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
limit(f(x),x,0);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: subsect start ]
What's Up? (and Down?)
   [wxMaxima: subsect end   ] */

/* [wxMaxima: comment start ]
We can get more insight into what is happening to the function 
(x(1-cos(x)))/(x-sin(x)) as x gets close to 0 by looking at the 
numerator and denominator of the function. In the following cell, 
we plot the numerator and denominator as separate functions of x.

We set the viewing window range so that the relation between the 
values of the numerator and denominator can be easily seen. You can 
zoom in closer by letting xb and yb get smaller and smaller. Your 
bounds for y will have to be much, much smaller than your bounds for
x; otherwise you will not be able to see your graphs.
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
num: x*(1-cos(x))$
den: x-sin(x)$
xb: 0.1$
yb: 10^(-4)$
wxdraw2d(
    nticks=200,
    color=red,
    explicit(num,x,-xb,xb),
    color=blue,
    explicit(den, x, -xb,xb),
    xrange=[-xb,xb],
    yrange=[-yb,yb],
    xlabel="x",
    ylabel="y");
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
Which curve do you think is the graph of the numerator function, and
which is the denominator? Why? See if you can differentiate between 
the two curves as x gets closer to 0.

Use the graphs above to find an approximate numeric relation between
the value of the numerator and the value of the denominator for x 
values that are close to 0. Based upon the relationship you find, 
explain why the value of the limit as x approaches 0 of 
(x(1-cos(x))/(x-sin(x)) is not surprising.

Try to generate a table with values of x, num, den, and num/dem in 
each row, for values of x close to 0. Comment on what is happening 
to the values of num, den and num/dem as x gets closer to 0.
   [wxMaxima: comment end   ] */

/* [wxMaxima: subsect start ]
Try Another 0/0 Expression
   [wxMaxima: subsect end   ] */

/* [wxMaxima: comment start ]
Find the limit as x approaches 0 of the function log(x)/(1-2^(x-1)).
Start by trying to evaluate the function at x = 1.
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
f (x) := log(x)/(1-2^(x-1));
f(1);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
We had better graph the function to see what is happening.
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
wxdraw2d(
    nticks=200,
    color=red,
    explicit(f(x),x,0.5,1.5),
    xlabel="x",
    ylabel="y");
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
How can you determine the precise limit? Try looking at some values 
of the function for x near 1.
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
count: 0$
for i: 0.9 thru 1.1 step 0.01 do block(
    if (is(equal(i,1))) then undefined: true else undefined: false,
    if (undefined) then mm[count]: [i, "undefined"]
    else mm[count]: [i, f(i)],
    count: count+1)$
matrix1: matrix(["x","f(x)"])$
for i: 1 thru (count-1) do block(
    matrix1: addrow(matrix1, mm[i]))$
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
print(matrix1)$
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
To investigate this limit, try some explorations like those in 
in Part I.
   [wxMaxima: comment end   ] */

/* [wxMaxima: section start ]
PART II: What is 0^0?
   [wxMaxima: section end   ] */

/* [wxMaxima: comment start ]
How does a function like |x|^x behave as x gets closer and closer to
0? The answer is not obvious, since numbers to the 0 power got to 1,
but 0 to any power is 0. We will start by trying the limit command.
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
kill(all)$
load(draw)$
ratprint: false$
numer:true$
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
g (x) := abs(x)^x;
limit(g(x),x,0);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
Is this correct? We might look at the graph of the funciton. Note 
what happens when x is 0 on the graph.
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
wxdraw2d(
    nticks=200,
    color=red,
    explicit(g(x),x,-2,2),
    xrange=[-2,2],
    yrange=[0,4],
    xlabel="x",
    ylabel="g(x)");
/* [wxMaxima: input   end   ] */

/* [wxMaxima: subsect start ]
Continuity Issues
   [wxMaxima: subsect end   ] */

/* [wxMaxima: comment start ]
If we ask for g(0), the result should indicate that it is an 
indeterminate form (0^0 is such a form).
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
g(0);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
The graph of g(x) appears to be continuous at x=0. Is it? Why?
   [wxMaxima: comment end   ] */

/* [wxMaxima: section start ]
PART III: One Sided Limits
   [wxMaxima: section end   ] */

/* [wxMaxima: comment start ]
Section 2.4, Exercise 18
   [wxMaxima: comment end   ] */

/* [wxMaxima: comment start ]
Consider the function (sqrt(2*x)*(x-1))/abs(x-1). This function 
appears to have problems at x = 1. Let's begin by looking at the 
graph.
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
h (x) := sqrt(2*x)*(x-1)/abs(x-1)$
wxdraw2d(
    nticks=200,
    color=red,
    explicit(h(x),x,0,3),
    xlabel="x",
    ylabel="y");
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
The vertical line at x = 1 really shouldn't be there. The lower 
piece is not really connected to the upper piece. Maxima graphs a 
function by plotting a series of points on the funciton and then 
connects them with straight lines, sometimes when they really 
shouldn't be connected.

What is happening at x = 1? Is h(x) defined at x = 1?
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
h(1);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
Lets' try the limit() command. Becasue of the input at x=1, we will 
use one-sided limits. The option minus indicates that we are 
approaching the value specified from values that are smaller, 
whereas plus indicates that we are approaching the value specified 
from values that are larger.
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
numer:false$
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
limit(sqrt(2*x)*(x-1)/(-(x-1)), x, 1, minus);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
limit(sqrt(2*x)*(x-1)/((x-1)), x, 1, plus);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
This result verifies what is shown in the graph. That is the limit
as x approachs 1(-) of h(x) is -sqrt(2) and the limit as x 
approachs 1(+) of h(x) is sqrt(2). We can check this out by looking 
at some values of the function x = 1 and comparing these values with
sqrt(2).
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
count: 1$
for i: 0.999 thru 1.001 step 0.0001 do block(
    if (is(equal(i,1))) then undefined: true else undefined: false,
    if (undefined) then mm[count]: [i, "undefined"]
    else mm[count]: [i, h(i)],
    count: count+1)$
matrix2: matrix(["x","h(x)"])$
for i: 1 thru (count-1) do block(
    matrix2: addrow(matrix2, mm[i]))$
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
print(matrix2)$
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
The value of sqrt(2) is 1.41421.
   [wxMaxima: comment end   ] */

/* [wxMaxima: subsect start ]
You Try It: Part III
   [wxMaxima: subsect end   ] */

/* [wxMaxima: comment start ]
Find the limit as x -> 0(+) of x*log(x).
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
g (x) := x*log((x))$
g(0);
wxdraw2d(
    nticks=200,
    color=red,
    explicit(g(x),x,-1,2));
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
What do you think is happening around x = 0? How do we know for sure
that there is not a jump discontinuity at x = 0? Maybe, if we zoom 
in close enough to x = 0, we will see a jump. How can we tell for 
sure whether there is or is not a jump at x = 0?
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
limit(g(x),x,1,plus);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: section start ]
PART IV:  What a Difference a Power Makes!
   [wxMaxima: section end   ] */

/* [wxMaxima: comment start ]
In your coursework, you have undoubtedly studied the limit of the 
sin(x)/x as x->0 and have found that its value is 1. What happens to
the limit if we change the power of x in the denominator to values 
that are near but not equal to 1? We begin by graphing three of 
these functions and see what the graphs suggest. The function 
sin(x)/x will be graphed in red, sin(x)/(x^0.9) in green, and 
sin(x)/(x^1.1) in blue. Because the roots of negative numbers are 
not always real numbers, we use a right-hand limit, approaching 0 
only from the right.
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
kill(all)$
load(draw)$
ratprint:false$
numer:true$
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
f1 (x) := sin(x)/x$
f2 (x) := sin(x)/(x^0.9)$
f3 (x) := sin(x)/(x^1.1)$
wxdraw2d(
    nticks=200,
    color=red,
    explicit(f1(x),x,0,5),
    color=green,
    explicit(f2(x),x,0,5),
    color=blue,
    explicit(f3(x),x,0,5));
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
Based on this graph, what would you guess are the limits of the 
three functions as x->0+?

Now let's evaluate each of the functions at x = 10^(-100).
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
value: 10^-100;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
f1(value);
f2(value);
f3(value);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
Do these results confirm or contradict the conjectures you made for 
the values of the three limits?
   [wxMaxima: comment end   ] */

/* [wxMaxima: subsect start ]
You Try It: PART IV
   [wxMaxima: subsect end   ] */

/* [wxMaxima: comment start ]
PROOFS ARE IMPORTANT!

Closely examine the sin(x)/(x^n) problem.

The following commands will result in the sine function being 
plotted in the thicker form in black, x in red, x^0.9 in green and 
x^1.1 in blue.
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
num: sin(x)$
den1: x$
den2: x^0.9$
den3: x^1.1$
b: 10^(-5)$
wxdraw2d(
    nticks=200,
    color=black,
    line_width=3,
    explicit(num,x,0,b),
    color=red,
    line_width=1,
    explicit(den1,x,0,b),
    color=green,
    explicit(den2,x,0,b),
    color=blue,
    explicit(den3,x,0,b));
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
What's going on here? Without being more careful, we might conclude 
from the graphs above that in all three cases the ratio of num to 
denom is a constant and that each limit should be approaching some 
finite number. To see what is really happening, you need to look at 
several graphs of the numerator and denominator functions, each one 
zoomed by a factor of 1/100. Go back to the preceding cell and 
re-execute the commands with the exponent of b = -7, -9, -11, . . .
What does this tell you about the ratio of num to denom in each of 
the three limits as x approaches 0?

Wait a minute! How do we know for sure that the pattern illustrated 
in the graphs above will continue if we zoom in even closer. Maybe 
things will "settle down" and the ratios will, in fact, approach 
constant values. What about the function (x*(1-cos(x))/(x-sin(x))
from Part I?  Maybe, if we zoom in close enough to 0, the graph of 
num to denom will start to look like the correspondin graphs for 
one of the three functions considered in this Part. How can we be 
sure of the situation? The answer is in the formal definition of 
the limit and proof. As mathematicians, we are not fully satisfied 
until we can prove our conjectures about the behavior of the 
functions as x approaches 0. Then, there is no doubt.
   [wxMaxima: comment end   ] */

/* Maxima can't load/batch files which end with a comment! */
"Created with wxMaxima"$