Generating Tables to Demonstrate Limits
Demonstrates the limit of a function f
by generating tables of
values of f
) for values of x
How to Use
How to Use
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
- Enter the function f (x) in the text input field
marked "f (x)="
- Enter the value of c in the text input field marked
- Enter the starting value of δ in the text input field
- 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 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
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.