1. Section 1.2: Finding Limits Graphically and Numerically Day 1
AP Calculus AB

2. Math Talk
Which is greater 24∙6 or 26∙4?

3. Presentation Key Colors
RED: important and key to write down
GREEN: important to write down, but you’ll need to summarize in own words. Also could be further example problems that you may or may not need. This is more up to your own discretion
BLUE: do not have to write down at all. Just used to illustrate a concept we’ll be discussing or exploring an idea before we formalize it

4. By the end of this lesson, you should …
Define components of limit notation
Define a limit
Determine the limit using a table
Determine the limit using a graph
Explain the difference between a limit and a function value
Identify and explain where limits do not exist
Draw graphs that show different limit/non-limit criteria

5. Limit Notation
Example: “What is the limit of 2𝑥+4 as 𝑥 approaches −6?

6. Informal Limit Definition
Example:
lim 𝑥→1 𝑥 3 −1 𝑥−1 =3
“The limit of 𝑥 3 −1 𝑥−1 as x approaches 1 is 3.”
Graphical:
LIMIT
Verbal:
lim 𝑥→𝑐 𝑓(𝑥) =𝐿
If 𝑓(𝑥) becomes arbitrarily close to L as x approaches c from either side, then the limit as x approaches c is L.
In your own words:

7. Graphical Approach
Find lim 𝑥→−2 𝑓(𝑥) for the following functions using their respective graphs
EX 1: EX 2: EX 3:

8. Graphical Approach EX 1: EX 2: EX 3: lim 𝑥→−2 𝑓 𝑥 =−1 lim 𝑥→−2 𝑓 𝑥 =−5
𝑓 −2 =1
𝑓 −2 =−5
𝑓 −2 =𝐸𝑟𝑟𝑜𝑟

9. Numerical Approach
Find lim 𝑥→0 𝑓(𝑥) for the following functions using their respective tables
EX 1:
EX 2:
EX 3: x
-0.01
-0.001
0.0001
0.001
0.01
F(x)
Error x
-0.01
-0.001
0.0001
0.001
0.01
F(x) -7 x
-0.01
-0.001
0.0001
0.001
0.01
F(x) -3

10. Numerical Approach EX 1: EX 2: EX 3: lim 𝑥→0 𝑓 𝑥 =2 𝑓 0 =Error
-0.01
-0.001
0.0001
0.001
0.01
F(x)
Error
lim 𝑥→0 𝑓 𝑥 =2
𝑓 0 =Error x
-0.01
-0.001
0.0001
0.001
0.01
F(x) -7
lim 𝑥→0 𝑓 𝑥 =15
𝑓 0 =−7 x
-0.01
-0.001
0.0001
0.001
0.01
F(x) -3
lim 𝑥→0 𝑓 𝑥 =−3
𝑓 0 =−3

11. Calculator Shortcuts: Using a Table to Find the Limit
Using the table to help find the limit of a function
Example: lim 𝑥→4 𝑥 2 −16 = ?
Step 1: Put 𝑓 𝑥 = 𝑥 2 −16 into “y =_____”
Step 2: Click “2nd” “Window/Tblset”
“Indpnt”: should be on “auto”
“∆Tbl”: should be a very small change in x (ex: .0001)
“TblStart”: near the value you are examining (ex: 4)
Step 3: Click “2nd” “Graph/Table”
Step 4: Analyze Results lim 𝑥→4 𝑥 2 −16 =0

12. Summary of Finding Limits Conceptually
In your own words, take 5 minutes to explain the following for your notes
1. How to find the limit using a table
Look at numbers VERY close to what value you are looking for on both sides
2. How to find the limit using a graph (this is really key!)
Use a vertical line to cover up the value you are looking for
Examine where both sides of the graph are approaching
3. What is the difference between a limit and a function value? How do they related to each other?
Limit: what occurs around a particular x-value
Function Value: where the function is defined at a particular x-value
Function value does not affect the limit at that point

13. Limits that Fail to Exist
Case 1: Behavior That Differs from Right and Left
Example: lim 𝑥→0 𝑥 𝑥
Notice that the red lines
are not meeting at one point
Also, at x = 0 is the only place
where the limit does not exist.
At any other point in the
function, the limit does exist.

14. Limits that Fail to Exist
Case 2: Unbounded Behavior
(some type of ±∞)
Example: lim 𝑥→ 𝑥 2
Once again, only at x = 0, the
limit does not exist. It does
exist every where else.
Key: ±∞ is not a value, thus
a function cannot be bounded
by it.

15. Limits that Fail to Exist
Case 3: Oscillating Behavior
Example: lim 𝑥→0 sin 1 𝑥
No matter how much you zoom in to x = 0,
the function keeps bouncing between -1 and 1.
Thus, we cannot conclude a limit.

16. Exit Ticket For Feedback
Refer to the graph 𝑓(𝑥)
1. lim 𝑥→ −5 𝑓(𝑥) =
2. lim 𝑥→2 𝑓(𝑥) =
3. 𝑓 −5 =
4. At what values of x does
the limit not exist?
𝑓(𝑥)