Estimation and Limits -- A Graphic Approach

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.

We are interested in

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

and we begin by drawing a graph of the function y = f(x). Next we mark the point a on the x-axis and the point L on the y-axis. In this particular picture notice that L = f(a) but this is not always true. In fact, usually we are interested in the limit because we can't actually compute f(a).

Now suppose that the permissible error for f(x) is epsilon. We mark the points L - epsilon and L + epsilon on the y-axis and we color the region between these two points green to indicate that this is the permissible region for f(x).

Finally, we look at the x-axis and mark two points a - delta and a + delta that are close enough to a so that if x is between these two points then f(x) will be in the permissible region. The region between a - delta and a + delta is colored yellow in the figure. Notice that if x is in the yellow region then f(x) is in the green region.

Next mark the points L - epsilon and L + epsilon on the y-axis and color the region between these two points green to indicate that it is the permissible region for f(x).

Finally mark points a - delta and a + delta on the x-axis choosing delta small enough so that if x is between these two points then f(x) is in the permissible region. Color the region between a - delta and a + delta yellow. Your figure should look generally like the figure above when you are done.

1.  
     Lim   2x + 4  = 10,      epsilon = 1
   x --> 3
   

2.  
     Lim   2x + 4  = 10,      epsilon = 0.5
   x --> 3
   

3.  
     Lim   3x + 4  = 7,      epsilon = 1
   x --> 1
   

4.  
     Lim   3x + 4  = 7,      epsilon = 0.5
   x --> 1
   

5.  
     Lim   -2x + 4  = 2,      epsilon = 1
   x --> 1
   

6.  
     Lim   -2x + 4  = 2,      epsilon = 0.5
   x --> 1
   

7. Suppose that f(x) = m x + b is a linear function. What can you say about the relationship between epsilon and delta?

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   For each of the remaining problems do exactly the same thing that you did for the first six problems above BUT use your CAS window to draw a graph of the function y = f(x), then print the graph, and complete the problem by hand on the printed graph.

8.  
     Lim   x^2  = 4,      epsilon = 1
   x --> 2
   

9.  
     Lim   x^2  = 4,      epsilon = 0.5
   x --> 2
   

10.  
      Lim   sin x = 0,      epsilon = 1
    x --> 0
    

11.  
      Lim   sin x = 0,      epsilon = 0.5
    x --> 0
    

12.  
      Lim   sin x = 1,      epsilon = 1
    x --> pi/2
    

13.  
      Lim   sin x = 1,      epsilon = 0.5
    x --> pi/2