Numerical Methods for finding
Limits
Goal: (1) To be able to interpret limit behavior based on looking at a table
of values.
What follows is a way of expressing what we
learned in the last lesson…
Informal Definition of a Limit (page 53)
If f (x) becomes arbitrarily close to a unique number L as
x approaches c from either side, then we say.... the limit
of f (x) as x approaches c is L.
We write it as... lim x→c f (x) = L
In the last lesson, we were determining limits by reading the graph.
We will learn a second technique to determine limits in this lesson, the
numeric method.
1. Let’s start with using the “check
the graph” technique
Put the equation in your calculator: careful with the
parentheses.
Put in the following window. The key with the
graphing calculator is getting the right viewing
window. It comes with practice.
Hit the graph key. Now ride the roller coaster
towards x = 0 from both directions.
Note: when x equals 0, there is no y-value
given as 0 is not in the domain of the function.
Now, let’s find the limit by checking
values close to x = 0
To for us to put in values for x go to TABLE
SETUP (2nd window) and make Indpnt ASK
Now go to TABLE (2nd graph) and put in small values
TO THE LEFT of 0
Now, let’s find the limit by checking
values close to x = 0
To get a better approximation of the y-value, move the
cursor to that y-value
Note: in the table to the right, the
calculator rounded the answer to 2 for an
approximation good to 11 decimal places.
Now, let’s do a right side limit
You can remove values from the table by
repeatedly hitting the delete button
By the Limit Existence Theorem, since the right
and left hand limits equal the same number, the
overall limit as x approaches 0 is 2.
Note: The fact that f(0) does
not exist (DNE) had no bearing
on the limit as x approaches 0!
Use both the GRAPH and the
TABLE to find the following
limit…..
Fill out the following table to show your numerical
evaluation of the limit as x approaches 2
Use both the GRAPH and
Numerical method to find the
following limit…..
Let’s use the Graph method first
Let’s focus in on the part of the graph near
x = 2
Use both the GRAPH and
Numerical method to find the
following limit…..
We can “blow up” the part of the graph we are interested in using
the ZOOM BOX (ZBOX) command.
The Limit appears to be approximately .25
Now, lets use the numerical
method utilizing a table of
values.
First, from the left-side….
x→2
−
From the left (less than 2) the values get closer to .25
Now, lets use the numerical
method utilizing a table of
values.
Now, from the right…..
x→2
+
x
2.001 2.01 2.1
f (x) .250 .249 .244
From the right (greater than 2) the values get closer to .25
You do not need an equation to use the numerical method
IF you have a table of values.
What is the limit provided by the table below (correct to 3 decimal places)?
f (x) 18.75 18.7346 18.73428
x
3.1
3.001
3.00001
18.73421
3.0000001
lim+ f (x) = 18.734
x→ 3
What limit(s) are provided by the table below (again, correct to 3
decimal places)?
f (x) 14.3989 14.4447 15.3162 15.3439
x
2.6
2.8
lim− f (x) = 15.344 and
x→ 3
2.99
2.999
lim f (x) Does Not Exist
x→ 3