Post on 03-Jan-2016
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.