Problem:
Investigate the behavior of the function

f(x) = sin(1/x)

for values of x near 0.


Visualization:

Using Microcalc:

  1. Choose Beginning Calculus from the initial menu and press .
  2. Use the arrow keys to move to the menu item Limit of F(x), press and .
  3. At the prompt F(x) =, enter the function
    sin(1/x)
  4. At the prompt a = , enter 0 .
  5. At the prompt h = , enter 1 .
  6. You will then see on the screen F(a+h) = 0.84147 098
  7. Press and use the arrow keys to move to h <<-- 1/2 h and press .
  8. If you keep repeating the previous step then you will obtain a table as below:
    __________________________________________________________
    
        N          a + h/2^n       F(a + h/2^n)
       ----------------------------------------
        0        1.00000 000        0.84147 098
        1        0.50000 000        0.90929 743
        2        0.25000 000       -0.75680 250
        3        0.12500 000        0.98935 825
        4        0.06250 000       -0.28790 332
        5        0.03125 000        0.55142 668
        6        0.01562 500        0.92002 604
        7   7.81250 000 e-03        0.72103 771
        8   3.90625 000 e-03       -0.99920 803
        9   1.95312 500 e-03        0.07951 849
       10   9.76562 500 e-04       -0.15853 338
    
    On the evidence of all values of  F  calculated so far,
    we estimate  Lim  F(x) =  -2.08027 731
                 x-ťa+
    
    Press :
    ________________________________________________________
    
  9. If you want to see the behavior for x < 0, press after seeing the values above, use the arrow keys to move to Change h and press .
  10. At the prompt h = , type -1 and proceed as above.