Example 1

 For example, let's explore   .  We first define    and notice that      is not defined at    .

>

restart:

>	f:=x->(x^3-x-6)/(x-2);

>

f(2);

However, if we plot  there seems to be no problem near .

>	plot(f(x),x=-1..3);

     This leads us to believe that the limit does  exist at .  For further evidence of this we compute some of the values of  when  is near  but not equal to it.

>

f(1.9);

>

f(1.99);

>	f(1.9999);

>

f(2.1);

>

f(2.01);

>	f(2.0001);

It becomes apparent from this output that as  gets closer to   ,     is getting closer to 11.  To make this clearer, we can tabulate     for many values of .  (The absolute value is given by the command: abs().)

     First we look at some -values which are less than 2:

>	abs(f(1.9)-11);

>	abs(f(1.99)-11);

>	abs(f(1.9999)-11);

Then we look at some -values greater than 2:

>	abs(f(2.1)-11);

>	abs(f(2.01)-11);

>	abs(f(2.0001)-11);

      When evaluating at  values of  which are very close to , we may need to increase the accuracy of Maple computation.  This is done with the Digits command.   For example, if we compute  with the standard eight digit accuracy we get a different value than if we use 20 digit accuracy:

>	f(2.00001);

>	Digits:=20:

>	f(2.00001);

Do Loops and Spreadsheets

        If we wish to get a bit fancy about tabulating these values we can use do-loops or spreadsheets to help organize the data.  Here are two do-loops that can be copied for use in other situations.

>

restart:

>	Digits:=25:

>	f:=x->(x^3-x-6)/(x-2);

>

a:=2;

>	for i from 1 to 10 do 'f'(a +(0.1)^i)=f(a +(0.1)^i)  end do;

>

>	for i from 1 to 10 do 'f'(a -(0.1)^i)=f(a -(0.1)^i)  end do;

     And here is a spreadsheet that can be copied for use with other functions.  To copy this spreadsheet somewhere else, first set your cursor where you want the new spreadsheet and then go to Insert on the toolbar and scrolldown to Spreadsheet.   This inserts a blank spreadsheet at the position of your cursor.   Next, select the spreadsheet below by clicking between the left sidebar and the spreadsheet, so that a box outlines the spreadsheet.  Then select Copy  from  the Edit  menu. Finally, return to the new spreadsheet that you inserted and select it so that it has a box around it, and then select Paste  from the Edit  menu.  After this, you may change   and    as you please and then use   Evaluate Spreadsheet   from the Spreadsheet  menu.  

       To find out more about creating your own spreadsheets with Maple look at the project called  "Limits with Spreadsheets".

>

restart:

>	Digits:=25:

>	f:=x->(x^3-x-6)/(x-2);

>

a:=2;

More Examples

  Let us now use Maple to explore .

After restarting Maple, we define the function.

>

restart:

>	f:=x->(x^3-2*x-6)/((x-1)^2);

Setting  we cut and past the do -loops  from above to explore the values of  near 1.

>

a:=1;

>	for i from 1 to 10 do 'f'(a+(0.1)^i)=f(a +(0.1)^i)  end do;

>	for i from 1 to 10 do 'f'(a-(0.1)^i)=f(a -(0.1)^i)  end do;

      Since these  are increasingly large negative numbers,  we are led to believe that the limit does not exist.  A graph  of this function helps to confirm this result:

>	plot(f(x),x=-1..3,y=-100..5);

As another example, let's look at .   We restart Maple and define   and  .

>

restart:

>	f:=x->sin(1/x);

>

a:=0;

>	for i from 1 to 10 do 'f'(a+(0.1)^i)=f(a +(0.1)^i)  end do;

>	for i from 1 to 10 do 'f'(a-(0.1)^i)=f(a -(0.1)^i)  end do;

Here we notice that the values of the function do not seem to "settle down" to any particular number, so again, we are led to believe that the limit fails to exist.   Again, let's look at the function to see what is happening near zero:

>	plot(f(x), x=-5..5);

     Finally, let's look at .

>

restart:

>	f:=x->sin(x)/x;

>

a:=0;

>	for i from 1 to 10 do 'f'(a+(0.1)^i)=f(a +(0.1)^i)  end do;

>	for i from 1 to 10 do 'f'(a-(0.1)^i)=f(a -(0.1)^i)  end do;

This data certainly leads us to believe that the limit exists and is 1.   Again, we can get a pictorial view of this with a graph:

>	plot(f(x), x=-10..10);

The   limit Command

    Maple has the power to evaluate limits for you using the limit  command.   Here is an example

>

restart:

>	f:=x->(x^3-x-6)/(x-2);

>	limit(f(x),x=2);

    This tells you that .   Of course it does not tell you how it got this answer, so if your Calculus instructor wants you to show your work when evaluating limits, this won't do that for you.  However, it does give a nice check that the answer you got is correct.

    In using the limit command, you must be careful about understanding the output.  For instance, in your Calculus class it might be the case that   is considered to be undefined, but Maple returns .

>	limit(1/x^2,x=0);

Of course, in a more sophisticated sense, this limit exists and is infinite.   On the other hand, Maple correctly states that   does not exist.

>	limit(1/x,x=0);

     Even more complicated output from Maple  can  come up.  This usually means that the limit does not exist in the standard sense, but that there may be a more sophisticated way of understanding the limiting value(s) of the function.     A particularly interesting example of this is given by .

>	limit(sin(1/x),x=0);

This particular output means that the limit does not  exist, but that values of  can be found arbitrarily close to any number between -1 and 1 when  is close to .