1. Limits
Lesson 3.1

2. What Is Happening?

3. Limit of a Function Given a function f(x)
Let a and L be real numbers
As x gets closer to a (but not equal to a)
Corresponding value of f(x) gets closer to L
The value of f(x) can be made as close to L as desired
Just choose a value of x close enough to a
• f(x) L
x • a

4. Limit of a Function
We say that L is the limit of f(x) as x approaches a
Note that the limit definition just says what happens when x gets close to a
The function might not be defined at a

5. Example Given function Yet we are interested in Define on Y= screen
What is the domain of the function?
Yet we are interested in
Define on Y= screen
Check values in the Tables (♦Y)
What value would you pick for L?
View Geogebra Demo

6. Other Possibilities What if Again the domain excludes +3
Enter the function
Check the Tables
What do you conclude for
View Geogebra Demo

7. Non-Existence of Limits
Limit will not exist when
f(x) becomes infinitely large in magnitude as x → a
Can be large, same sign on both sides of a
Can be and on opposite sides of a
Possible that and
Then no limit exists

8. Properties of Limits Note Rules for Limits, pg 155 Important concepts
Limit of sum or difference is sum or difference of the limits
Similar for products, quotients (where denominator not zero)
If p(x) is a polynomial then

9. Limits at Infinity
Recall our study of end behavior of a rational function
The long run (end) behavior is determined by the quotient of the leading terms
Leading term dominates for large values of x for polynomial
Leading terms dominate for the quotient for extreme x

10. Note how the end behavior depends on the leading coefficients
Limits at Infinity
Note how the end behavior depends on the leading coefficients
Given
Note the value for a large x

11. Limits at Infinity
Thus, for We say

12. Try It Out
Consider the following …

13. Assignment
Lesson 3.1
Page 163
Exercises 1 – 51 odd