1. 1.2 Finding Limits Graphically and Numerically, part 1
OBJECTIVE
Estimate a limit using a numerical or graphical approach.
Learn different ways that a limit can fail to exist.

2. Consider this function.
What happens to the value of f (x) when the value of x gets closer and closer and closer (but not necessarily equal) to 2?
Read as the limit of f(x) as x approaches 2.
We write this as:
The answer can be found graphically, numerically, and analytically.

3. Consider this function.
What happens to the value of f (x) when the value of x gets closer and closer and closer (but not necessarily equal) to 2?
Read as the limit of f(x) as x approaches 2.

4. “the limit as x approaches a of f(x) is L.”
DEFINITION
As x approaches a, the f(x) approaches a certain value, L, called the limit.
“the limit as x approaches a of f(x) is L.”

5. Finding the limit numerically

6. 1) Consider the functionx
4.9
4.99
4.999
4.9999 5
5.0001
5.001
5.01
5.1
g(x)
0.9
0.99
0.999
0.9999 1
1.0001
1.001
1.01
1.1 1
Basic idea: plug and chug and typically you’ll end up with a finite number.
However, most limit problems don’t work out with plug and chug.
Often you’ll get 0/0 or nonzero/0
So you need different techniques for finding the limit.
Typically you’ll end up with a finite number

7. 2) Consider the functionx
1.9
1.99
1.999
1.9999 2
2.0001
2.001
2.01
2.1
f(x)
3.9
3.99
3.999
3.9999
und.
4.0001
4.001
4.01
4.1

8. 3) Consider the functionx
1.9
1.99
1.999
1.9999 2
2.0001
2.001
2.01
2.1
f(x)
11.41
11.94
11.994
11.999
und.
12.001
12.006
12.06
12.61

9. 4) Consider the functionx -4 -3 -2 -1 1 2 3 4
h(x) -1 -1 -1 -1
und. 1 1 1 1
One sided limits are in section 1.4
does not exist
The roads don’t meet, so the limit d.n.e.

10. “the limit as x approaches 0 from the left of h(x) is -1.”
One-side limits
For a limit to exist, the function must approach the same value from both sides.
One-sided limits approach from either the left or right side only.
“the limit as x approaches 0 from the left of h(x) is -1.”

11. = 2 2 2 1 5) left hand limit right hand limit value of the function
Also the limit as x approaches 1 has nothing to do with the value of f at 1.
You can fill the circle, or move the closed circle and the limit is still going to be 2.
At any point where the graph of f is continuous the y-coordinate will equal the limit of f as x approaches the x-coordinate at that point. So the value and limit coincide where ever the graph of f is continuous
value of the function 1

12. 2 1 2 6) does not exist left hand limit right hand limit
The roads don’t meet, so the limit d.n.e. 6)
does not exist
left hand limit 2
right hand limit 1
There is no limit of f as x approaches 0. The lim as x →0 does not exist.
However, we can describe the behavior of f near x = 0 in terms of one-sided limits.
value of the function 2

13. 1 1 2 7) = 1 left hand limit right hand limit value of the function
There is no limit of f as x approaches 0. The lim as x →0 does not exist.
However, we can describe the behavior of f near x = 0 in terms of one-sided limits.
value of the function 2

14. 1 1 8) d.n.e. left hand limit right hand limit value of the function
The roads don’t meet, so the limit d.n.e. 8)
d.n.e.
left hand limit
right hand limit 1
There is no limit of f as x approaches 0. The lim as x →0 does not exist.
However, we can describe the behavior of f near x = 0 in terms of one-sided limits.
value of the function 1

15. What are your questions?

16. Student Questionnaire (graded)
Homework
Read the syllabus
Student Questionnaire (graded)
Worksheet. Limits Worksheet A