Example: f (x)=x2
Enter the value of c in the text input field marked "c=".
Enter the starting value of d in the text input field marked "d=".
Click the "Restart" button to generate the tables.
Click the "Closer (2x)" button to move the values shown in the tables to values closer to c (by a factor of 2.0).
To generate a new table after entering new values or regenerate the table from the starting values, click the "Restart" button.
To erase the tables and set the text input fields to default values, click the "Clear" button.
Difference Quotient: (ex+1-e)/x as x approaches c=0 -- limit is derivative of ex at x=1
Jump Discontinuity: |x|/x as x approaches c=0 -- note differing one-sided limits
Asymptote: 1/x as x approaches c=0 -- note diverging values in both tables
sin(x)/x as x approaches c=0
from the right
(uses conditional expression to delete domain to left)
Recall the "-" definition of a limit:
The limit of f (x) as x approaches c
is equal to L if and only if:
for every there is a so that
|f (x)-L|< whenever |x-c|<.
In the tables generated by the applet, the values of d roughly correspond to the values of in the definition -- the limit of f (x) as x approaches c is L just in case sufficiently small values of d in the tables give values of f (x) sufficiently close to the value of L.
If the values in either table diverge, then the limit does not exist. Also, if the values in both tables approach limits, but the limits do not agree, then the limit does not exist; however, it may be possible to demonstrate a one-sided limit in this case.