1. Ways to Evaluate Limits:
Graphically – show graph and arrows traveling from each side of the x- value to find limit
Numerically – show table values from both the right and left of the x- value to discover limit
Analytically - algebraically

2. In AP Calculus, we will be approaching problems in three different ways:
Analytically (using the equation)
Numerically
Graphically

3. Finding Limits Graphically and Numerically

4. Finding Limits Graphically The informal definition of a limit is “what is happening to y as x gets close to a certain number”

5. Notation for a Limit
This is read: The limit of f of x as x approaches c equals L.

6. If we are concerned with the limit of f(x) as we approach some value c from the left hand side, we write

7. One-sided Limits—Left-Hand Limit
This is read--The limit of f as x approaches c from the left.

8. If we are concerned with the limit of f(x) as we approach some value c from the right hand side, we write

9. One-sided Limits—Right-Hand Limit
This is read: The limit of f as x approaches c from the right.

10. Definition of a Limit
If the right-hand and left-hand limits are equal and exist, then the limit exists.

11. In order for a limit to exist at c=
and we write:

12. Example 2--Graphically
Look at the graph and notice that y approaches 2 as x approaches 1 from the left. This is also true from the right. Therefore the limit exists and is 2.

13. Graphically!
What causes this discontinuity???
ALGEBRA!

14. Simplify:

15. When we are computing limits the question that we are really asking is what y value is our graph “intending to take” as we move on towards x = 2 on our graph. We are NOT asking what y value the graph takes at the point in question!

16. Example 2--Numerically
Look at the table and notice that y approaches 2 as x approaches 1 from the left.(slightly smaller than 1) This is also true from the right. (slightly larger than 1) Therefore the limit Exists and is 2.

17. Finding Limits
EXAMPLE
Determine whether the limit exists. If it does, compute it.
SOLUTION
Let us make a table of values of x approaching 4 and the corresponding values of x3 – 7 as we approach for both from above (from the right) and below (from the left)
x x3 - 7
As x approaches 4, it appears that x3 – 7 approaches 57. In terms of our notation,

18. One more try….. Turn on the TI-83/84 or 89 Graph y = 2x + 2
Create a table
Study what happens as x approaches 5.
Make sure your tblset is set to: Independent “ask”. You can then choose any x value you like and get its y-value

19. From the Left
From the Right x
f(x) x
f(x)

20. Example. Evaluate the following limit:
Let’s make a table of values…

21. From the Left
From the Right x
f(x) x
f(x)

22. Graphically!
What causes this discontinuity???
ALGEBRA!

23. Simplify:

24. When we are computing limits the question that we are really asking is what y value is our graph “intending to take” as we move on towards x = 2 on our graph. We are NOT asking what y value the graph takes at the point in question!

25. Example. Evaluate the following limit:

26. The limit is NOT 5!!! Remember from the discussion after the first example that limits do not care what the function is actually doing at the point in question. Limits are only concerned with what is going on around the point.
Since the only thing about the function that we actually changed was its behavior at x = 2 this will not change the limit.

27. Finding Limits
EXAMPLE
For the following function g (x), determine whether or not exists. If so, give the limit.
SOLUTION
We can see that as x gets closer and closer to 3, the values of g(x) get closer and closer to 2. This is true for values of x to both the right and the left of 3.

28. Limit of the Function Note: we can approach a limit from
left … right …both sides
Function may or may not exist at that point
At a
right hand limit, no left
function not defined
At b
left handed limit, no right
function defined a b

29. Observing a Limit
Can be observed on a graph.

30. Observing a Limit
Can be observed on a graph.

31. Find each limit, if it exists.
In order for a limit to exist, the two sides of a graph must match at the given x-value.
D.S.

32. 1.1 Limits: A Numerical and Graphical Approach
Thus for Example 1:
does not exist

33. Non Existent Limits
f(x) grows without bound

34. Non Existent Limits

35. What happens as x approaches zero?
Graphical Example 2
What happens as x approaches zero?
The limit as x approaches zero does not exist.

36. What happens as x approaches zero?
Graphical Example 2
What happens as x approaches zero?
The limit as x approaches zero does not exist.

37. From this graph we can see that as we move in towards t=0 the function starts oscillating wildly and in fact the oscillations increases in speed the closer to t=0 that we get.  Recall from our definition of the limit that in order for a limit to exist the function must be settling down in towards a single value as we get closer to the point in question.
This function clearly does not settle in towards a single number and so this limit does not exist!

38. Common Types of Behavior Associated with Nonexistence of a Limit

39. SUMMATION Introduction to limits
The limit of a function is the y value the graph is getting closer to as x gets closer to a particular value
Making a table of values to calculate the limit – must be done on a calculator
Sketch a graph to calculate the limit, or use an already existing graph to calculate the limit

40. When limits fail to exist
When the right hand and left hand limits do not agree
When there is unbounded behavior
(as we have just seen)
3. When there is oscillating behavior