1. Rates of Change and Limits
2.1
Rates of Change and Limits

2. Quick Review
In Exercises 1 – 4, find f (2).

3. Quick Review
In Exercises 5 – 8, write the inequality in the form a < x < b.

4. Quick Review
In Exercises 9 and 10, write the fraction in reduced form.

5. What you’ll learn about
Average and Instantaneous Speed
Definition of Limit
Properties of Limits
One-Sided and Two-Sided Limits
Sandwich Theorem
Essential Question
How can limits be used to describe continuity, the derivative
and the integral: the ideas giving the foundation of
Calculus?

6. Average and Instantaneous Speed
A body’s average speed during an interval of time is found by dividing the distance covered by the elapsed time.
Experiments show that a dense solid object dropped from rest to fall freely near the surface of the earth will fall y = 16t 2 feet in the first t seconds.
Wile E Coyote drops an anvil from the top of a cliff. What is its average rate of speed during the first 5 seconds of its fall?

7. Average and Instantaneous Speed
A body’s average speed during an interval of time is found by dividing the distance covered by the elapsed time.
Experiments show that a dense solid object dropped from rest to fall freely near the surface of the earth will fall y = 16t 2 feet in the first t seconds.
What is the speed of the anvil at the instant t = 5?
Since we cannot calculate the speed right at 5 sec. Calculate the average between 5 sec and slightly after 5 sec.
The smaller h becomes the closer the average will get to 160.

8. Definition of Limit
Let c and L be real numbers. The function f has limit L as x approaches c if, given any positive number e, there is a positive number d such that for all x,
We write
The sentence is read, “The limit of f of x as x approaches c equals L.” The notation means that the values of x approach (but does not equal) c.
The next figures illustrate the fact that the existence of a limit as x → c never depend on how the function may not be defined at c.

9. Definition of Limit continued
The function f has a limit 2 as x → 1 even though f is not defined at 1.
The function g has a limit 2 as x → 1 even though g(1) ≠ 2.
The function h is the only one whose limit as x → 1 equals its value at x = 1.

10. Properties of Limits

11. Properties of Limits continued
Product Rule:
Constant Multiple Rule:

12. Properties of Limits continued

13. Example Properties of Limits
Use any properties of limits to find:
Sum and difference rules
Product and multiple rules
Quotient rule
Sum and difference rules
Product and multiple rules

14. Polynomial and Rational Functions
Example Limits
Determine the limit by substitution. Support graphically.

15. Evaluating Limits Example Limits
As with polynomials, limits of many familiar functions
can be found by substitution at points where they are
defined. This includes trigonometric functions,
exponential and logarithmic functions, and composites
of these functions.
You can evaluate limits either graphically, numerically, or algebraically.
Example Limits
Determine the limit by substitution. Support graphically.

16. Example Limits Solve graphically: Confirm algebraically:
Determine the limit graphically. Support algebraically.
Solve graphically:
Confirm algebraically:

17. Example Limits Limit does not exist. x y Solve graphically:
Determine the limit graphically. Support algebraically.
Solve graphically:
Confirm algebraically:
Confirm numerically: x y
Limit does not exist.

18. Example Limits Solve graphically: Confirm algebraically:
Determine the limit graphically. Support algebraically.
Solve graphically:
Confirm algebraically:

19. Example Limits
Determine the limit graphically. Support algebraically.
Solve graphically:
Confirm algebraically:
On page 60, the graph and table shows that this limit approaches 1.

20. One-Sided and Two-Sided Limits

21. Example One-Sided and Two-Sided Limits
Solve graphically:
Solve graphically:

22. Example One-Sided and Two-Sided Limits
Given the following graph, compute the limits.

23. Example One-Sided and Two-Sided Limits
Given the following graph, compute the limits.

24. Example One-Sided and Two-Sided Limits
Given the following graph, compute the limits.

25. Sandwich Theorem

26. Pg. 66, 2.1 #1-47 odd