1. 1.2 Finding Limits
Numerically and Graphically

2. Limits
A function f(x) has a limit L as x approaches c if we can get f(x) as close to c as possible but not equal to c.
x is very close to, not necessarily at, a certain number c
NOTATION:

3. 3 Ways to find Limits
Numerically - construct a table of values and move arbitrarily close to c
Graphically - exam the behavior of graph close to the c
Analytically

4. 1) Given , find 2 x
1.9
1.99
1.999
1.9999
3.61
3.9601 4 2 x
2.0001
2.001
2.01
2.1
4.0401
4.41 4

5. 2) Given , find 1 x
0.9
0.99
0.999
0.9999
2.710
2.9701 3 1 x
1.0001
1.001
1.01
1.1
3.0301
3.31 3

6. 3. What does the following table suggest aboutx
0.9
0.99
0.999
1.001
1.01
1.1
F(x) 7 25
4317
3.0001
3.0047
3.01

7. Finding Limits Graphically
There is a hole in the graph.
Limits that Exist even though the function fails to Exist

8. One sided Limits
notation
Limits from the right
Limits from the left

9. 4) Use the graph of to find

10. 5) Use the graph of to find

11. 6) Finding Limits graphically
a) Find
b) Find
c) Find
d) Find
e) Find

12. 7) Use the graph of to find1 –1
Does Not Exist – DNE

13. Limits that Fail to Exist
In order for a limit to exist the limit must be the same from both the left and right sides. 1 –1

14. Limits that Fail to Exist
The behavior is unbounded or approaches an asymptote 1 –1

15. Limits that Fail to Exist
The behavior oscillates

16. HOMEWORK
Page 54 # 1-10 all numerically # 11 – 26 all graphically