1.1

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Limits: A Numerical and

Graphical Approach

OBJECTIVE

•Find limits of functions, if they exist, using

numerical or graphical methods.

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Question: As the input x gets “closer” to 3, what happens

to the value of 𝑓(𝑥)?

• 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 “closer” to 3, 𝑓(𝑥) 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 “as x gets ‘closer’ to the number a, 𝑓(𝑥) gets closer to the number L”

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 𝑓(𝑎) does not matter (𝑓 𝑎 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 𝐻(𝑥) graphed below

Question: If 𝑥 is less than 1, but gets

closer to 1, what happens to the value

of 𝐻(𝑥)?

Answer: 𝐻(𝑥) 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 𝐻(𝑥) graphed below

Question: If 𝑥 is greater than 1, but gets

closer to 1, what happens to the value

of 𝐻(𝑥)?

Answer: 𝐻(𝑥) gets closer to −2

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. lim𝑥→𝑎

𝑓 𝑥 = 𝐿 if and only if

lim𝑥→𝑎−

𝑓 𝑥 = 𝐿 and lim𝑥→𝑎+

𝑓 𝑥 = 𝐿

2. If lim𝑥→𝑎−

𝑓 𝑥 ≠ lim𝑥→𝑎+

𝑓 𝑥 , then lim𝑥→𝑎

𝑓 𝑥 = 𝐿 does not

exist

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Example: Consider the function 𝐻(𝑥) graphed below

Question: Does lim𝑥→1

𝑓 𝑥 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 𝐻(𝑥) graphed below

Question: Does lim𝑥→−3

𝑓 𝑥 exist?

Answer: Yes, lim𝑥→−3

𝑓 𝑥 = −4

1.1 Limits: A Numerical and Graphical Approach

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1.1 Limits: A Numerical and Graphical Approach

Example: Calculate the following limits based on the

graph of 𝑓

a.)

b.)

c.)

2lim ( )x

f x

2lim ( )x

f x

2lim ( )x

f x

= 3

= 3

= 3

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= 0 = 1

= 1 = 1

= 4 = 2

Does not exist = 1

= 1 = 1

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Example: Consider the function 𝑓 𝑥 =1

𝑥

Find the following limits:

a) lim𝑥→0−

𝑓 𝑥

b) lim𝑥→0+

𝑓 𝑥

c) lim𝑥→0

𝑓 𝑥

graphically and numerically

1.1 Limits: A Numerical and Graphical Approach

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Solutions:

a) lim𝑥→0−

𝑓 𝑥 = −∞

Means: As 𝑥 gets closer to 0 from

the left, 𝑓(𝑥) gets more negative

b) lim𝑥→0+

𝑓 𝑥 = ∞

Means: As 𝑥 gets closer to 0 from

the right, 𝑓(𝑥) gets more positive

c) lim𝑥→0

𝑓 𝑥 Does not exist

1.1 Limits: A Numerical and Graphical Approach

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Important Point: ∞ is not a number!

• A number is a destination on the number line

• ∞ is not a destination, it is a journey

1.1 Limits: A Numerical and Graphical Approach

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Example: Consider the function 𝑓 𝑥 =1

𝑥

Find the following limits:

a) lim𝑥→−∞

𝑓 𝑥

b) lim𝑥→∞

𝑓 𝑥

graphically and numerically

1.1 Limits: A Numerical and Graphical Approach

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Solutions:

a) lim𝑥→−∞

𝑓 𝑥 = 0

Means: As 𝑥 gets more negative,

𝑓(𝑥) gets closer to 0

b) lim𝑥→∞

𝑓 𝑥 = 0

Means: As 𝑥 gets more positive,

𝑓(𝑥) 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

• 1.1 HW

1.1 Limits: A Numerical and Graphical Approach