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2.2 Finding Limits Graphically and Numerically
A limit is where ________________________________
x 0.75 0.9 0.99 0.999 1 1.001 1.01 1.1 1.25
f (x)
Fill in the blanks on the table using
Warm-up:
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Limit notation
Other ways to describe limits
· intended ___________________________________________
· look for ___________ based on _________________________
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WHAT IS A LIMIT???????????
http://www.calculus-help.com/tutorials/
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As f(x) gets closer to 3, the graph is.....
If there is a hole in the graph.... ___________________
1
2
3
4
5 6 7 8 9 10
1
2 3 4
5
6
7
8
9
10
x
y
So the
Example 1: Refer to the graph below to answer the questions on the right.
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Three ways to find a limit:
1. Substitution
2. Factoring
3. Graphically
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Limits by Substitution
Example 2: Find each limit using substitution.
a.
b.
c.
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Limits by Factoring
Example 3: Find each limit using factoring.
a.
b.
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Definition of a Limit
The function has limit 2 as even though is not defined at 1.
The function is the only one whose limit as equals its value at .
The function has limit 2 as even though .
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One and Two-sided Limits
Right-hand: is the limit of as approaches from the right.
Left-hand: is the limit of as approaches from the left.
A function has a limit as approaches if and only if the right-hand and left-hand limits at exist and are equal. In symbols,
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Whether you are approaching from the left side or right side, you will get closer and closer to 1.
1
2
3
4
5 6 7 8 9 10
1
2 3 4
5
6
7
8
9
10
x
y
If
then...
Example 4: Refer to the graph below to answer the question on the right.
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Example 5: Refer to the graph below to find the limits.
a.
b.
c.
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Example 6: Refer to the graph below to find the limits.
a.
b.
c.
d.
e.
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Piecewise Functions Limits
Example 7: Graph the piecewise function and find the limits.
a.
b.
c.
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Properties of LimitsIf are real numbers and
then:
(The limit of a constant is a constant.)
(i.e., Direct substitution)
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Find
Practice: Find each limit.
b.
c.
a.
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