The Limit Does Not Exist II

Section 1.1.6

Last time

• We learned that a limit does not exist for a certain x when the left and right limits do not equal.

This time

• We are going more in depth.

• We will learn about other situations in which the limit does not exist.

Learning Objectives

• Given a function and a point at which to evaluate the limit, determine the existence of the limit using the criteria of– Left-right limit equivalence– Endpoint rule– Oscillation rule

Example 1

• Does the limit exist as x3? Why or why not?

Example 2

• Find lim f(x) as x0 when f(x) = abs(2x)/x.

Asymptotic Limits

• Asymptote: a line to which a function eternally gets closer and closer.

• There are both vertical and horizontal asymptotes.

• Let’s stick with vertical for now.

• Observe the function f(x) = 1/x2 on the right. What do you think is its limit as x0?

• Let’s look at f(x) = -1/x2. What is the limit as x0?

Example 3

• Now let’s look at f(x) = 1/x. What is the limit as x0?

Where Edison Online and I Disagree

• Edison Online states, both in the lesson and textbook, that the limit does not exist as x approaches any vertical asymptote.

• As you saw in previous slides, I disagree. The limit is ∞ or -∞ as long as both the left and right limits equal.

• I have emailed Mr. Rideout about this.• I doubt that you will need to worry about

this disagreement when taking your quiz.

Endpoint Functions (Ex. 4)

• Some functions have an endpoint, such as f(x) = sqrt(x).

• What do you think is the limit of f(x) = sqrt(x) as x0? Why?

Background Sin/Cos Knowledge

sin 0 = 0

sin π = 0

sin 2π = 0

sin 3π = 0

sin 4π = 0

sin 5π = 0

and so on…

cos 0 = 1

cos π = -1

cos 2π = 1

cos 3π = -1

cos 4π = 1

cos 5π = -1

and so on…

Oscillation

• Oscillate: to go back and forth between two numbers.

• The function below oscillates between -1 and 1 as x0.

Example 4

• Find the limit of cos (1/x) as x0 using a table of values.

You can do this without a calculator. Let’s see how…

• We know that cos π = -1.• With f(x) = cos (1/x), if we plug 1/π in for x, we

will get cos π.

1

1111

x

• Realize that 1/π, 1/2π, 1/3π, etc. get closer to 0.

• Therefore, our table of values becomes…

-1/π -1/2π -1/3π 0 1/3π 1/2π 1/π

- -1

Don’t forget: cos(-x) = cos(x)

Wrap-Up

• Asymptotic limits

• The limit does not exist as xc when– The left and right limits don’t equal– c is an endpoint.– The function oscillates as xc.

Homework

• Textbook 1.1.6 #1-4 (all problems).

If you can’t get the textbook to work, please do the reteaching problems.