Warm-Up Thoughts(after 1.2)• Why do piece-wise functions and rational functions make for great “limit” examples and discussion. Think of at least 3 reasons and be ready to share out.

• If f(2) = 4, can you conclude anything about the limit of f(x) as x approaches 2? Explain your reasoning.

• If the limit of f(x) as x approaches 2 is 4, can you conclude anything about f(2)? Explain your reasoning.

How was HW #1-26 Section 1.2• Did you check your ODD answers in the back of the

book?• I have EVEN answers on the back whiteboard.• Most of the time, I will post the scanned answer key online

on my website since it shows step-by-step but I wait until after our first day of discussion to do this. I want to hear from you! What went wrong? Why?

Syllabus• Let’s talk through this today.

Reflection on Reading of 1.2

Limits that Fail To Exist #1 Behavior That Differs from the right and left. Sketch a graphical example of this.

#2 Unbounded Behavior: the limit is Sketch a graphical example of this AND write a rational function that behaves this way.

#3 Oscillating Behavior. Sketch a graphical example of this.

SECTION 1.3 EVALUATING LIMITS ANALYTICALLY

Question for You…

• Does the limit of f(x) as x approaches c depend on the value of the function at x = c? Use a picture to help answer the question.

• Sometimes, the limit is exactly the value of the function such the Can you think of when this might occur and why.

Are You Behaving For Me?

Since most of our functions are “well behaved” and continuous (watch out for discontinuous functions), a limit can be found by direct substitution.

Why? Why? Why?

Let b and c be real numbers and let n be a positive integer.

1. What “family of functions” is this?

2. What “family of functions” is this?

3. What “family of functions” is this?

Theorem 1.1 Some Basic Limits

Theorem 1.2 Properties of Limits

• See page 59 of the text

• Let us talk through each of these carefully and think about why they are true statements.

The Limit of a Polynomial

• Can direct substitution be a quick approach to finding a limit of a polynomial? Why or why not?

• Evaluate

(Do this 2 ways: 1st with the properties of limits and the 2nd with direct substitution)

Limit of a Rational Function• Can direct substitution be a quick approach to finding a

limit of a rational? Why or why not?

With rationals, what do you always need to be watchful of?

(Do this 2 ways: 1st with the properties of limits and the 2nd with direct substitution)

The Limit of a Radical Function

• Can direct substitution be a quick approach to finding a limit of a radical function? Consider an even root versus and odd root. Why or why not?

The Limit of a Composite Function

• What is a composite function?

• Try…

• Try…

• This theorem only works if each of the unique functions involved in the composition are continuous functions. Always watch out and be thinking!

Limits of Trig Functions

Let c be a real number in the domain of the given trig. function, then the following is true:

1.

When would direct substitution fail?

Practice: Think about what property you are applying.

1. ?

What’s This?• Evaluate using direct substitution (our most favorite way)

What is wrong? Why is this a problem? What tools do you have that would enable you to find the actual limit?

Indeterminate Form for our course, will only cover the form or and when you use direct substitution and this is your end result, the limit can’t be determined from this form alone so we have to use some algebraic tricks (later on some derivative tricks too).

Indeterminate Form Continued• Evaluate using direct substitution (our most favorite way)

• What happened?• Evaluate using direct substitution (our most favorite way)

lim𝑡→4

𝑡−√3 𝑡+44−𝑡

Two Special Trig Limits You must memorize these!!!

1.

Ex 1:

Ex 2:

BONUS: 1 arguments must match!

Is the same true for

Challenge (if time permits)

Closure: Strategy for Finding Limits• Learn to recognize which limits will allow direct

substitution.• Apply the properties such as composite, radical, rational,

sum and difference, scalar, quotient, etc.• Use a graph• Use a table via the calculator• Algebraic techniques (simplify the fraction, rationalize the

numerator, use a conjugate as a fancy 1, use a trig. identity…just get creative)

Homework• Section 1.3 #1-81 multiples of 3

(since we will have another worksheet to review indeterminate form)

• Remember: Your job is to isolate the problems you can’t do and ask me in class the next day!

• READ Section 1.3 if you need more clarification. We will cover the Squeeze Theorem a little bit later.