1. Definition and finding the limit
When substitution results in a 0/0 fraction, the result is called an indeterminate form.
The limit of an indeterminate form exists, but to find it you must use a technique, such as factor and cancel.

2. In class: worksheets ( possibilities Worksheet limits or McCafrey)
Calculus Date: 9/26/14
Objective: SWBAT define, calculate & apply properties of limits graphically, numerically and now analytically.
Do Now –
Mini Quiz 5 minutes Take out a piece of paper. Can be a half sheet.
HW Requests: HW: pg 30 SM all
In class: worksheets ( possibilities Worksheet limits or McCafrey)
HW: Complete Worksheets
Announcements:
Mandatory session Sine and Cosine functions starting
with the Unit Circle
Quiz Friday
To get ahead,
You have to do extra!

3. Show your work then 2.
Mini Quiz
7 minutes

4. Techniques-Finding limits for Rational Expressions
Try Substitution, if doesn’t work
Try Factor and cancel and then
3. Try Substitution again, if doesn’t work
4. Do DNE or +/- infinity check
- If the right and left side limit are not equal the limit does not exist - DNE
Let’s go to the SM pg #28 #1-8 (10) HW: pg 30 SM all

5. If the right and left side limit are not equal the limit does not exist – DNE
One sided Limits
If the left side number is negative then the lim 𝑥→ 𝑐 − 𝑓 𝑥 =−∞
If the left side number is positive then the lim 𝑥→ 𝑐 − 𝑓 𝑥 =∞
If the right side number is negative then the lim 𝑥→ 𝑐 + 𝑓 𝑥 =−∞
If the left side number is negative then the lim 𝑥→ 𝑐 + 𝑓 𝑥 =∞

6. Rationalizing Technique
If there is a radical in the numerator or the denominator,
rationalize, simplify and cancel, then try substitution.
Substituting we get
Hint: Often you can cancel a common term in the
numerator and denominator when simplifying

7. Rationalizing Technique
Rationalize, simplify (cancel) and try substitution.
Substituting we get = 1 4

8. f(0)is undefined; 2 is the limit
Try This
Find: 2
f(0)is undefined; 2 is the limit

9. f(0) is defined; 2 is the limit
Find:
Try This
f(0) is defined; 2 is the limit 2 1
1, x = 0

10. Try This
Find the limit if it exists:
DNE

11. Try This
Find: if

12. Try This
Find the limit of f(x) as x approaches 3 where f is defined by:

13. Example
Find the limit if it exists:
Try substitution

14. Example Find the limit if it exists:
Substitution doesn’t work…does this mean the limit doesn’t exist?

15. Use the factor & cancellation technique
and
are the same except at x=-1

16. Use the factor & cancellation technique
After factoring and cancelling, now try substituting -1 again.
= 3

17. Try This Isn’t that easy? Find the limit if it exists:
Did you think calculus was going to be difficult? 5

18. Try This
Solve using limit properties and substitution: 6

19. Try This
Find the limit if it exists:

20. Example Sometimes limits do not exist. Consider:
If substitution gives a constant divided by 0, the limit does not exist (DNE)

21. Try This Find the limit if it exists: Confirm by graphing
The limit doesn’t exist

22. Lesson Close
Name 3 ways a limit may fail to exist.

23. Exit Ticket
In Class: SM – pg 28 #1-5
HW: SM pg 30 #1-15

24. Try This
Find the limit if it exists: -5

25. Limit properties again
The existence or non-existence of f(x) as x approaches c has no bearing on the existence of the limit of f(x) as x approaches c.
What matters is…what value does f(x) get very, very close to as x gets very, very close to c. This value is the limit.

26. Watch out for piecewise functions
Limits, again!
In order for a limit to exist at c, the left-hand limit must equal the right hand limit.
If the left-hand limit equals the right hand limit, then the limit exists and we write:
Watch out for piecewise functions

27. When finding the limit of a function it is important to let x approach a from both the right and left. If the same value of L is approached by the function then the limit exist and

28. Consider
Example
for
and =?

29. Try This
Graph and find the limit (if it exists):
DNE

30. Example
Trig functions may have limits.

31. Try This

32. Using the Product Rule Technique

33. Important Idea
The functions have the same limit as x-1

34. Try This Graph and on the same
axes. What is the difference between these graphs?

35. Analysis
Why is there a “hole” in the graph at x=1?

36. Example Consider What happens at x=1?
Let x get close to 1 from the left: x
.75 .9
.99
.999
f(x)

37. Try This Consider Let x get close to 1 from the right: x 1.25 1.1 1.01
1.001
f(x)

38. Try This
What number does f(x) approach as x approaches 1 from the left and from the right?

39. Informal Definition of Limit
If f(x) becomes arbitrarily close to a single number L as x approaches a number c from either side, the limit of f(x), as x approaches c, is L.

40. - Definition of Limit
Let f be a function defined on an open interval containing c (except possibly at c) and let L be a real number. The statement means that for each >0 there exists a >0 such that if then .

41. Basic Limits

42. Constant Function Limits
a and b are both constants
This means that for any constant function f(x) = b, as x approaches any constant a, the limit will always be b.

43. Linear Function Limits
The limit of f(x) = x as x approaches any constant is the constant itself.

44. Exponential Function Limits
Just plug in a for x

45. Properties Let and Scalar multiple: Sum or difference: Product:
Quotient: , if K0
Power:

46. Let's try a practice problem.
Property (B) tells us we can split these apart:
Using limit (1) and limit (2) from the basic limits, we get:

47. Putting it all together
So,
This is called the Substitution method

48. Try This
Solve using limit properties and substitution: 25