MAT 1234 Calculus I

1. MAT 1234Calculus I Section 1.5 The Limit of a Function http://myhome.spu.edu/lauw

2. WebAssign • Homework 1.5 • Be sure to read the instructions.

3. What do we care? • How fast “things” are going • The velocity of a particle • The “speed” of formation of chemicals • The rate of change of charges in a capacitor

4. Recall • Limit of the following form is important

5. Recall

6. Recall

7. Recall • Limit of the following form is important • 1.4: Estimate limits by tables • 1.6: Compute limits by algebra • 1.5: Formally define limits

8. Preview • One-sided limits • The existence of limits • Infinite limits

9. One-Sided Limits • We will focus on the conceptual meaning and understanding. • We will not look closely at the formal definition.

10. Left-Hand Limit y • The left-hand limit is 2 when x approaches 3 • Notation: • Independent of f(3) 2 y=f(x) x 3

11. Left-Hand Limit y 2 y=f(x) x 3

12. Left-Hand Limit y 2 y=f(x) x 3

13. Left-Hand Limit y 2 y=f(x) x 3

14. Left-Hand Limit (Formal)

15. Right-Hand Limit y • The right-hand limit is 4 when x approaches 2 • Notation: • Independent of f(2) 4 y=f(x) x 2

16. Right-Hand Limit y 4 y=f(x) x 2

17. Right-Hand Limit y 4 y=f(x) x 2

18. Right-Hand Limit y 4 y=f(x) x 2

19. Right-Hand Limit (Formal)

20. Limit of a Function

21. Example 1 y 1.5 x 2

22. Example 2 y 2.5 1.5 x 2

23. Infinite Limits y • The left-hand limit DNE • Notation: y=f(x) is not a number x a

24. Infinite Limits • Let us look at all possible situations and their standard representations.

25. Infinite Limits y • The left-hand limit DNE • Notation: x a

26. Infinite Limits y • The right-hand limit DNE • Notation: x a

27. Infinite Limits y • The right-hand limit DNE • Notation: x a

28. Infinite Limits y • The limit DNE • Notation: x a

29. Infinite Limits y • The limit DNE • Notation: x a

30. Infinite Limits y • The limit DNE • Notation: x a

31. Review: We learned… • left-hand and right-hand limits • definition of limits • infinite limits • infinity is a concept, not a number • to use infinity to represent geometry concepts - unboundness

32. Classwork • Use pencils! • Work with your partner ONLY. • No discussions between groups (penalty = last group to get out) • You can go, after Tara or I check both of your work.