limits: a numerical and graphical approachestrada.cune.edu/facweb/brian.albright/sect11.pdfthe left,...
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1.1
2012 Pearson Education, Inc.
All rights reserved Slide 1.1-1
Limits: A Numerical and
Graphical Approach
OBJECTIVE
โขFind limits of functions, if they exist, using
numerical or graphical methods.
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Question: As the input x gets โcloserโ to 3, what happens
to the value of ๐(๐ฅ)?
โข Analyze it graphically and numerically
1.1 Limits: A Numerical and Graphical Approach
2
( )3
9xf x
x
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Answer: As x gets โcloserโ to 3, ๐(๐ฅ) gets closer to 6
We write
1.1 Limits: A Numerical and Graphical Approach
3lim ( ) 6x
f x
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DEFINITION (informal):
The expression
means โas x gets โcloserโ to the number a, ๐(๐ฅ) gets closer to the number Lโ
1.1 Limits: A Numerical and Graphical Approach
lim ( )x a
f x L
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Comments
1. x never reaches a
2. The value of ๐(๐) does not matter (๐ ๐ need not even
be defined)
3. x can approach from either direction, or both directions
1.1 Limits: A Numerical and Graphical Approach
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Example: Consider the function ๐ป(๐ฅ) graphed below
Question: If ๐ฅ is less than 1, but gets
closer to 1, what happens to the value
of ๐ป(๐ฅ)?
Answer: ๐ป(๐ฅ) gets closer to 4
We write:
Called a limit from the left
1.1 Limits: A Numerical and Graphical Approach
1lim ( ) 4x
H x
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Example: Consider the function ๐ป(๐ฅ) graphed below
Question: If ๐ฅ is greater than 1, but gets
closer to 1, what happens to the value
of ๐ป(๐ฅ)?
Answer: ๐ป(๐ฅ) gets closer to โ2
We write:
Called a limit from the right
1.1 Limits: A Numerical and Graphical Approach
1lim ( ) 1x
H x
1.1 Limits: A Numerical and Graphical Approach
THEOREMS
1. lim๐ฅโ๐
๐ ๐ฅ = ๐ฟ if and only if
lim๐ฅโ๐โ
๐ ๐ฅ = ๐ฟ and lim๐ฅโ๐+
๐ ๐ฅ = ๐ฟ
2. If lim๐ฅโ๐โ
๐ ๐ฅ โ lim๐ฅโ๐+
๐ ๐ฅ , then lim๐ฅโ๐
๐ ๐ฅ = ๐ฟ does not
exist
2012 Pearson Education, Inc. All rights reserved Slide 1.1-8
2012 Pearson Education, Inc. All rights reserved Slide 1.1-9
Example: Consider the function ๐ป(๐ฅ) graphed below
Question: Does lim๐ฅโ1
๐ ๐ฅ exist?
Answer: NO, since the limit
from the left does not equal the
limit from the right
1.1 Limits: A Numerical and Graphical Approach
1 1lim ( ) 4, lim ( ) 1x x
H x H x
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Example: Consider the function ๐ป(๐ฅ) graphed below
Question: Does lim๐ฅโโ3
๐ ๐ฅ exist?
Answer: Yes, lim๐ฅโโ3
๐ ๐ฅ = โ4
1.1 Limits: A Numerical and Graphical Approach
2012 Pearson Education, Inc. All rights reserved Slide 1.1-11
1.1 Limits: A Numerical and Graphical Approach
Example: Calculate the following limits based on the
graph of ๐
a.)
b.)
c.)
2lim ( )x
f x
2lim ( )x
f x
2lim ( )x
f x
= 3
= 3
= 3
Slide 1.1-12
= 0 = 1
= 1 = 1
= 4 = 2
Does not exist = 1
= 1 = 1
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Example: Consider the function ๐ ๐ฅ =1
๐ฅ
Find the following limits:
a) lim๐ฅโ0โ
๐ ๐ฅ
b) lim๐ฅโ0+
๐ ๐ฅ
c) lim๐ฅโ0
๐ ๐ฅ
graphically and numerically
1.1 Limits: A Numerical and Graphical Approach
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Solutions:
a) lim๐ฅโ0โ
๐ ๐ฅ = โโ
Means: As ๐ฅ gets closer to 0 from
the left, ๐(๐ฅ) gets more negative
b) lim๐ฅโ0+
๐ ๐ฅ = โ
Means: As ๐ฅ gets closer to 0 from
the right, ๐(๐ฅ) gets more positive
c) lim๐ฅโ0
๐ ๐ฅ Does not exist
1.1 Limits: A Numerical and Graphical Approach
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Important Point: โ is not a number!
โข A number is a destination on the number line
โข โ is not a destination, it is a journey
1.1 Limits: A Numerical and Graphical Approach
2012 Pearson Education, Inc. All rights reserved Slide 1.1-16
Example: Consider the function ๐ ๐ฅ =1
๐ฅ
Find the following limits:
a) lim๐ฅโโโ
๐ ๐ฅ
b) lim๐ฅโโ
๐ ๐ฅ
graphically and numerically
1.1 Limits: A Numerical and Graphical Approach
2012 Pearson Education, Inc. All rights reserved Slide 1.1-17
Solutions:
a) lim๐ฅโโโ
๐ ๐ฅ = 0
Means: As ๐ฅ gets more negative,
๐(๐ฅ) gets closer to 0
b) lim๐ฅโโ
๐ ๐ฅ = 0
Means: As ๐ฅ gets more positive,
๐(๐ฅ) gets closer to 0
1.1 Limits: A Numerical and Graphical Approach
Slide 1.1-18
= 1 = -1
Does not exist = 2
= 0 Does not exist
= 3 = 0
= 1 = 2
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= 3.30, 3.30, 3.30
= 2.90, 3.30, Does not exist
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Homework
โข 1.1 HW
1.1 Limits: A Numerical and Graphical Approach