1. 1. 1 A Preview of Calculus and 1
1.1 A Preview of Calculus and 1.2 Finding Limits Graphically and Numerically

2. Objectives
Understand what calculus is and how it compares to precalculus.
Estimate a limit using a numerical or graphical approach.
Learn different ways that a limit can fail to exist.

3. Swimming Speed See page 40 Questions 1-5
Swimming Speed: Taking it to the Limit
Questions 1-5

4. Preview of Calculus
Diagrams on pages 43 and 44

5. Two Areas of Calculus: Differentiation
Animation of Differentiation

6. Two Areas of Calculus: Integration
Animation of Integration

7. Limits
Both branches of calculus were originally explored using limits.
Limits help define calculus.

8. 1.2 Finding Limits Graphically and Numerically

9. Find the Limit x
.75 .9
.99
.999 1
1.001
1.01
1.1
1.25
f(x)
2.313
2.710
2.970
2.997 ?
3.003
3.03
3.310
3.813
x approaches 1 from the left
x approaches 1 from the right
Limits are independent of single points.

10. Exploration (p. 48) From the graph, it looks like f(2) is defined.
Look at the table.
On the calculator: tblstart 1.8 and ∆Tbl=0.1.
Look at the table again.
What does f approach as x gets closer to 2 from both sides?

11. Exploration (p. 48)
Look at the graph and the table.

12. Example
Limits are NOT affected by single points!

13. Three Examples of Limits that Fail to Exist
If the left-hand limit doesn't equal right-hand limit, the two-sided limit does not exist.

14. Three Examples of Limits that Fail to Exist
If the graph approaches ∞ or -∞ from one or both sides, the limit does not exist.

15. Three Examples of Limits that Fail to Exist
Look at the graph and table.
As x gets close to 0, f(x) doesn't approach a number, but oscillates back and forth.
If the graph has an oscillating behavior, the limit does not exist.

16. Limits that Fail to Exist
f(x) approaches a different number from the right side of c than it approaches from the left side.
f(x) increases or decreases without bound as x approaches c.
f(x) oscillates as x approaches c.

17. Homework
1.2 (page 54)
#5-17 odd