1.2 Finding Limits Graphically
Thursday, August 24, 2017
2:48 PM
1.2 Skills
• Evaluating limits with graphing calculator
○ Pretty math
○ Tables
• Three nonexistence of limit behaviors
• Removable vs nonremovable discontinuities
• Jump/skip discontinuities
You have a calculator…use it to calculate the following limits. Is there more than
one way to calculate a limit with a graphing calculator? Is there a best way?
Hint: Be aware of your MODE!
Given the graph of a function below. Compute each limit, if possible. If the limit does not exist (DNE),
please justify. In other words, what is it about the graph that makes the limit not exist?
Finally, what's the difference between the value of a function at a given x-value and the limit of a
function as you approach that x-value? Are they related? If one exists, must the other also?
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1.2 Review
Sunday, August 27, 2017
6:50 PM
1. What are the three different types of discontinuity? Describe or sketch an example
of each and name them, if possible.
1. There are three categories of functions for which a limit does not exist. Give an
example of each.
1. For the following problem, find the limit using your calculator.
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1.3 Finding Limits Analytically
Thursday, August 24, 2017
2:50 PM
What is it about a function that makes finding limits
potentially interesting?
1.3 Skills
• Evaluating limits analytically
○ Limit of a constant
○ Limits of continuous functions
○ Techniques for rational functions
• Special Trig limits
• Review: Unit Circle
With your partner, list as many continuous/potentially
discontinuous functions that you can think of.
Continuous functions
(potentially) discontinuous functions
Before we get into evaluating some limits, here are a couple interesting and basic theorems about limits:
Applying properties to evaluate a limit.
Chapter 1 Page 3
Now let's look at some examples of limits of discontinuous functions and learn some
tricks for evaluating them.
To evaluate trig limits involving fractions, we will want to use the identities we evaluated with our calculators in
section 1.2. Note: You will want to commit these to memory.
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Chapter 1 Page 5
Unit Circle
Tuesday, August 25, 2020
9:23 AM
Chapter 1 Page 6
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3.5 Limits at Infinity
Monday, August 19, 2019
9:34 PM
When asked to find a limit as x approaches positive/negative infinity, what are you
really being asked to find?
Chapter 1 Page 8
1.5 Infinite Limits
Friday, August 25, 2017
8:52 AM
1.5 Skills
• Finding equations for vertical asymptotes
• Limits at vertical asymptotes
• One-sided limits
Which functions (can) have vertical asymptotes?
How do we find them? What happens to a function as it approaches a vertical
asymptote? Does the limit exist? What if it's one-sided?
What's the difference between infinite limits and limits at infinity?
Chapter 1 Page 9
Intermediate Value Theorum
Monday, August 31, 2020
12:53 PM
If u have to pick up ur pencil
it's not continuous
Chapter 1 Page 10
1.4 Intermediate Value Theorem
Monday, August 28, 2017
11:17 AM
Unit circle
IVT
Piecewise functions on TI (67)
Reasons for discontinuity (3,6)
Making piecewise functions continuous (57, 60)
Intermediate Value Theorem
If a function is continuous on [a,b] and k is a number between f(a) and f(b), then there
exists a c in [a,b] such that f (c)=k.
Does IVT apply? If so, find the c-value guaranteed.
Chapter 1 Page 11
1.4 Continuity and One-Sided Limits
Thursday, August 24, 2017
2:29 PM
1.4 Skills
• Evaluating one-sided limits analytically
• Finding points of discontinuity
• Making piecewise functions continuous
• Definition of continuity
○ Graphic
○ Formal
What is the difference between "undefined" and "indeterminate"?
What are the three different types of discontinuity? Describe or sketch an example
of each and name them, if possible.
A function f is continuous at x = c if the following three conditions are met:
How do you find a point of discontinuity in a rational function?
What about a trig function?
Popular application of continuity
Chapter 1 Page 12
Notes on Quiz
Wednesday, September 9, 2020
7:27 AM
My Observations:
- DNE not LDE
- Having trouble with infinities
- Need to memorize limit rules
- Asymptotes with limits
Chapter 1 Page 13
What's on the test
Thursday, September 10, 2020
8:22 AM
- Anything on the quizzes
- Intermediate value theorem
- Definition of continuity
○ Be able to state it
○ Be able to tell which
functions aren't/are
continuous; be able to
show mathematically
- Vertical asymptotes:
○ Rational functions
○ Trig functions
- Step functions
○ Will be on the test just
like the quiz
- Making piecewise functions
continuous
- GO OVER AP CLASSROOM
Vertical asymptote with trig practice:
How to study
- Khan academy
- Limits practice
Chapter 1 Page 14