### Generating Tables to Demonstrate Limits

Demonstrates the limit of a function f (x) at x=c by generating tables of values of f (x) for values of x near c.
How to Use   ||   Examples   ||   Other Notes

How to Use
• Enter the function f (x) in the text input field marked "f (x)="
• Enter the value of c in the text input field marked "c="
• Enter the starting value of δ in the text input field marked "δ="
• 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)
• To regenerate the table from the starting values, generate a new table after entering new values, click the "Restart" button
The text input field marked "f (x)=" can accept a wide variety of expressions to represent functions. The text input fields marked "c=" and "δ=" can accept real numbers in decimal notation.
Examples

Other Notes
The starting value of δ determines where the tables start, with c-δ and f (c-δ) on the left, c+δ and f (c+δ) on the right. The other lines in the tables show values of f (x) for values closer to c -- on the left, c-δ/2, c-δ/4, c-δ/8, etc.

Recall the "ε-δ" definition of a limit:

The limit of f (x) as x approaches c is equal to L if and only if:
for every ε>0 there is a δ>0 so that |f (x)-L|<ε whenever |x-c|<δ

In the tables above, the values of δ roughly correspond to the values of δ in the definition -- the limit of f (x) as x approaches c is L just in case sufficiently small values of δ 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.