Copyright 2001
Department of Mathematics
University of Georgia
Athens, Georgia

John Gosselin

Some Interesting Limits

The family of functions (a+x)^(b+sin(1/x))

In this lab you will investigate the behavior of the function f(x) = (a+x)^(b+sin(1/x))  for various values of a, b and c as x  approaches 0 from the right and from the left. If you are unfamiliar with creating spreadsheets in Maple, refer to the introduction of the previous lab Limits and Spreadsheets in Maple .

Project

The function f(x) = sin(1/x)

We begin by studying the function f(x) = sin(1/x)  near x  = 0.

1. Make a plot of f( x ) on the interval [-1,1]. Describe the behavior of this function as x  approaches 0 from either side.

2. Find a number c1  with abs(c1) < 0 .001 such that f( c1 ) = 1/2  and another number c2  with abs(c2) < .1e-2  such that f(c2) = -1/2 . Does f( x ) have a limit as x  approaches 0?

3. Apply the limit command with x  approaching 0 to the function f( x ). Note that in some instances when the limit does not exist Maple returns a range of values in the form a.. b . This indicates that any number r  in the interval [ a,b ] is the limit of certain values of f( x ) with x  approaching 0. More precisely this means that one can find a sequence of numbers x[n]  approaching 0 and f( x[n] ) approaching r . Find a sequence of numbers x[n]  approaching 0 with f( x[n] ) approaching 1/2  and another sequence of numbers z[n]  approaching 0 with f( z[n] ) approaching -1/2 .

The case a  = b  =  1

>    restart:

Define the variables a  and b  and c  as 1. Also define the function f( x ).

1.Use a spreadsheet to display the values of f( x ) at  the sequence of points x[j] = 10^(-j) , j  = 1..10, approaching x  = 0 from the right. Does f( x ) appear to have a limit from the right?

2.Use a spreadsheet to display the values of f( x ) at the sequence of points x[j] = -10^(-j) , j  = 1..10, approaching x  = 0 from the left. Does f( x ) appear to have a limit from the left?

3. Does it appear that f( x ) has a limit as x  approaches 0? Explain.

4. Plot the function f( x ) on the interval from -0.5 to 0.5. Describe the behavior of the function on either side of zero. Does the graph indicate that f( x ) has a limit as x  approaches 0? If so, what does the limit appear to be?

5. Apply the limit command from Maple to f( x ) with x  approaching zero. Explain the result.

The case a  = 2 and b  = 3

>    restart;

Define the variables a  = 2 and b  = 3. Also define the function f( x ).

1. Use a spreadsheet to display the values of f( x ) at the sequence of points x[j] = 10^(-j) , j  = 1..10, approaching x  = 0 from the right. Does f( x ) appear to have a limit from the right?

2. Use a spreadsheet to display the values of f( x ) at the sequence of points x[j] = -10^(-j) , j  = 1..10, approaching x  = 0 from the left. Does f( x ) appear to have a limit from the left?

3. Does it appear that f( x ) has a limit as x  approaches 0? Explain.

4. Plot the function f( x ) on the interval from -0.5 to 0.5. Describe the behavior of the function on either side of zero. Does the graph indicate that f( x ) has a limit as x  approaches 0? If so, what does the limit appear to be?

5. Apply the limit command from Maple to f( x ) with x  approaching zero. Explain the result.

Extra Credit

 

>    restart:

Try some different values of positive values for a  and b  with f( x ) = (a+x)^(b+sin(1/x)) . Plot the functions on an interval centered at x  = 0. Use the limit command from Maple with x  approaching 0. Can you guess a formula for the output in terms of a  and b ? Try to justify your formula from your plots.

The Most Common Maple Commands

Academic Honesty Statement:

Place the following statement (by copying and pasting) at the end of your report and sign it in ink.  Your instructor will not grade your report unless this signed statement appears at the end of your report.

I understand that I may work with others if I give them credit in this statement.  I also understand that I am required to write my report--that to copy all or part of someone else's report or to allow someone else to copy all or part of my report constitutes plagiarism, which is a serious violation of academic honesty.

I worked with (replace this parenthetical remark with first and last names of those with whom you worked)  on this project.  I wrote my own report.  I did not copy any of this report from anyone else and I did not allow anyone else to copy any of this report.

Signed: