1. Rates of Change and Limits
AP CALCULUS AB
Chapter 2:
Limits and Continuity
Section 2.1:
Rates of Change and Limits

2. What you’ll learn about
Average and Instantaneous Speed
Definition of Limit
Properties of Limits
One-Sided and Two-Sided Limits
Sandwich Theorem
…and why
Limits can be used to describe continuity, the derivative and the integral: the ideas giving the foundation of calculus.

3. Average and Instantaneous Speed

4. Section 2.1 – Rates of Change and Limits
Instantaneous Speed:
To find the instantaneous speed, we start by calculating the average speed over an interval from time t1 to any slight time later t2=t1+h
To find the instantaneous speed, we take increasingly smaller values of h --- in other words, we let h approach 0.
Then

5. Definition of Limit
(If the horizontal distance between x and c is less than , then the vertical distance between and L is less than ).
or as x gets increasingly closer to c, then gets increasingly closer to L.

6. Definition of Limit continued

7. Definition of Limit continued

8. Properties of Limits

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

10. Properties of Limits continued

11. Example Properties of Limits

12. Polynomial and Rational Functions

13. Polynomial and Rational Functions

14. Example Limits

15. Section 2.1 – Rates of Change and Limits
Techniques for Finding Limits
Numerically – plug in values that approach c from both the right and left.
Algebraically – factor and cancel/simplify first. Then plug in c for x.
Graphically.
In “well-behaved” functions we can find the
by direct substitution of c for x:

16. Evaluating 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.

17. Section 2.1 – Rates of Change and Limits
Limits of Trigonometric Functions:

18. Example Limits

19. Example Limits
[-6,6] by [-10,10]

20. Section 2.1 – Rates of Change and Limits
Functions that agree in all but 1 point

21. Section 2.1 – Rates of Change and Limits
Cancellation Techniques for finding Limits
If you have a rational function

22. Section 2.1 – Rates of Change and Limits
Rationalization Techniques for Finding Limits

23. Section 2.1 – Rates of Change and Limits
Special Limits from Trigonometry

24. Section 2.1 – Rates of Change and Limits
Remember:

25. Section 2.1 – Rates of Change and Limits
As you approach 2 from the left, you get closer and closer to 12.
As you approach 2 from the right, you get closer and closer to 12.
So,

26. Section 2.1 – Rates of Change and Limits
Limit of a Composite Function:
If and
then

27. One-Sided and Two-Sided Limits

28. One-Sided and Two-Sided Limits continued

29. Section 2.1 – Rates of Change and LimitsIf
As x approaches 2 from the left, f(x) approaches 3.
As x approaches 2 from the right, f(x) approaches 3.
So,

30. Example One-Sided and Two-Sided Limits
Find the following limits from the given graph. 4 o 1 2 3

31. Section 2.1 – Rates of Change and Limits
Limits that do not exist:
As x approaches 0 from the left, f(x) approaches 0.
As x approaches 0 from the right, f(x) approaches 1.
So,
does not exist

32. Section 2.1 – Rates of Change and Limits
More Limits that do not exist:
As x approaches , from the left, f(x) goes to
As x approaches , from the right, f(x) goes to So
does not exist.

33. Section 2.1 – Rates of Change and Limits
More Limits that do not exist:
Oscillating behavior
Graph on calculator and zoom in about 4 times around x=0.

34. Sandwich Theorem

35. Sandwich Theorem

36. Section 2.1 – Rates of Change and Limits
The Sandwich Theorem:
If for all
In some interval about c, and
Then
g(x)
f(x)
h(x)