Numerical Methods for finding Limits

Goal: (1) To be able to interpret limit behavior based on looking at a table of values.

Informal Definition of a Limit (page 53)

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.

What follows is a way of expressing what we learned in the last lesson…

If f (x) becomes arbitrarily close to a unique number L asx approaches c from either side, then we say.... the limitof f (x) as x approaches c is L.

We write it as... lim x→c f (x) = L

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 focus in on the part of the graph near x = 2

Let’s use the Graph method first

Use both the GRAPH and Numerical method to find the following limit…..

The Limit appears to be approximately .25

We can “blow up” the part of the graph we are interested in using the ZOOM BOX (ZBOX) command.

Now, lets use the numerical method utilizing a table of values.

From the left (less than 2) the values get closer to .25

First, from the left-side…. x→ 2−

Now, lets use the numerical method utilizing a table of values.

From the right (greater than 2) the values get closer to .25

x 2.001 2.01 2.1f (x) .250 .249 .244

Now, from the right….. x→ 2+

f (x) 18.75 18.7346 18.73428 18.73421x 3.1 3.001 3.00001 3.0000001

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)?

limx→3+

f (x) = 18.734

What limit(s) are provided by the table below (again, correct to 3 decimal places)?

f (x) 14.3989 14.4447 15.3162 15.3439x 2.6 2.8 2.99 2.999

limx→3−

f (x) = 15.344 and limx→3

f (x) Does Not Exist