1. 11.1 Finding Limits Graphically and Numerically
2014 Limits Introduction
11.1
Copyright © Cengage Learning. All rights reserved.

2. Objectives Estimate a limit using a numerical or graphical approach.
Learn different ways that a limit can fail to exist.
Study and use the informal definition of limit.

3. Formal definition of a Limit:
If f(x) becomes arbitrarily close to a single number L as x approaches c from either side, the limit of f(x), as x approaches c, is L.
This limit is written as
“The limit of f of x as x approaches c is L.”

4. Properties of Limits:
Limits can be added, subtracted, multiplied, multiplied by a constant, divided, and raised to a power.
For a limit to exist, the function must approach the same value from both sides.
One-sided limits approach from either the left or right side only.

5. Limits can be found in various ways:
Graphically
Numerically
Algebraically
Ex: Find the following limit algebraically:

6. An Introduction to Limits
Ex: Find the following limit graphically:

7. Looks like y=1

8. An Introduction to Limits
Ex: Find the following limit numerically:

9. An Introduction to Limits
Start by sketching a graph of the function
For all values other than x = 1, you can use standard
curve-sketching techniques.
However, at x = 1, it is not clear what to expect.
We can find this limit numerically:

10. An Introduction to Limits
To get an idea of the behavior of the graph of f near x = 1, you can use two sets of x-values–one set that approaches 1 from the left and one set that approaches 1 from the right, as shown in the table.

11. An Introduction to Limits
The graph of f is a parabola that has a gap at the point (1, 3), as shown in the Figure 1.5.
Although x can not equal 1, you can move arbitrarily close to 1, and as a result f(x) moves arbitrarily close to 3.
Using limit notation, you can write
Figure 1.5
This is read as “the limit of f(x) as x approaches 1 is 3.”

12. This discussion leads to an informal definition of a limit:
A limit is the value (meaning y value) a function approaches as x approaches a particular value from the left and from the right.

13. The limit of a function refers to the value that the function approaches, not the actual value (if any).
not 1

14. DNE does not exist because the left and right hand limits do
not match! 2 1 1 2 3 4
At x=1:
left hand limit
right hand limit
value of the function
DNE

15. because the left and right hand limits match.2 1 1 2 3 4
At x=2:
left hand limit
right hand limit
value of the function

16. because the left and right hand limits match.2 1 1 2 3 4
At x=3:
left hand limit
right hand limit
value of the function

17. You Try– Estimating a Limit Numerically
Use the table feature of your graphing calculator to evaluate the function at several points near x = 0 and use the results to estimate the limit:

18. Example 1 – Solution
The table lists the values of f(x) for several x-values near 0.

19. Example 1 – Solution
cont’d
From the results shown in the table, you can estimate the limit to be 2.
This limit is reinforced by the graph of f (see Figure 1.6.)
Figure 1.6

20. Find
Graphic
Investigation

21. Use a graphing utility to estimate the limit:

22. Limits That Fail to Exist

23. Show that does not exist
Non-existance
Show that does not exist
Because the behavior differs from the right and from the left of zero, the limit DNE.

24. Discuss the existence of the limit:
Solution: Using a graphical representation, you can see that x does not approach any number. Therefore, the limit

25. The graph oscillates, so the limit does not exist.
Example 5:
Make a table approaching 0 x
x→0
sin 1/x 1 -1
DNE
The graph oscillates, so the limit does not exist.

26. Fig. 1.10, p. 51

27. Limits That Fail to Exist - 3 Reasons

28. Limits Basics
Examples

29. Properties of Limits Scalar multiple: Sum or difference: Product:
Quotient:
Power:
Direct substitution works for all polynomial and rational functions with non-zero denominators.

30. Using Properties of Limits

31. Find the following limits:

32. Group Work:
Sketch the graph of f. Identify the values of c for which exists.

33. Homework Day 1
MMM 25,26
Homework Day 2
Pg odd