1.1 2012 Pearson Education, Inc. All rights reserved Slide 1.1-1 Limits: A Numerical and Graphical Approach OBJECTIVE β’ Find limits of functions, if they exist, using numerical or graphical methods.
1.1
2012 Pearson Education, Inc.
All rights reserved Slide 1.1-1
Limits: A Numerical and
Graphical Approach
OBJECTIVE
β’Find limits of functions, if they exist, using
numerical or graphical methods.
2012 Pearson Education, Inc. All rights reserved Slide 1.1-2
Question: As the input x gets βcloserβ to 3, what happens
to the value of π(π₯)?
β’ Analyze it graphically and numerically
1.1 Limits: A Numerical and Graphical Approach
2
( )3
9xf x
x
2012 Pearson Education, Inc. All rights reserved Slide 1.1-3
Answer: As x gets βcloserβ to 3, π(π₯) gets closer to 6
We write
1.1 Limits: A Numerical and Graphical Approach
3lim ( ) 6x
f x
2012 Pearson Education, Inc. All rights reserved Slide 1.1-4
DEFINITION (informal):
The expression
means βas x gets βcloserβ to the number a, π(π₯) gets closer to the number Lβ
1.1 Limits: A Numerical and Graphical Approach
lim ( )x a
f x L
2012 Pearson Education, Inc. All rights reserved Slide 1.1-5
Comments
1. x never reaches a
2. The value of π(π) does not matter (π π need not even
be defined)
3. x can approach from either direction, or both directions
1.1 Limits: A Numerical and Graphical Approach
2012 Pearson Education, Inc. All rights reserved Slide 1.1-6
Example: Consider the function π»(π₯) graphed below
Question: If π₯ is less than 1, but gets
closer to 1, what happens to the value
of π»(π₯)?
Answer: π»(π₯) gets closer to 4
We write:
Called a limit from the left
1.1 Limits: A Numerical and Graphical Approach
1lim ( ) 4x
H x
2012 Pearson Education, Inc. All rights reserved Slide 1.1-7
Example: Consider the function π»(π₯) graphed below
Question: If π₯ is greater than 1, but gets
closer to 1, what happens to the value
of π»(π₯)?
Answer: π»(π₯) gets closer to β2
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. limπ₯βπ
π π₯ = πΏ if and only if
limπ₯βπβ
π π₯ = πΏ and limπ₯βπ+
π π₯ = πΏ
2. If limπ₯βπβ
π π₯ β limπ₯βπ+
π π₯ , then limπ₯βπ
π π₯ = πΏ 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 π»(π₯) graphed below
Question: Does limπ₯β1
π π₯ 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
2012 Pearson Education, Inc. All rights reserved Slide 1.1-10
Example: Consider the function π»(π₯) graphed below
Question: Does limπ₯ββ3
π π₯ exist?
Answer: Yes, limπ₯ββ3
π π₯ = β4
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 π
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
2012 Pearson Education, Inc. All rights reserved Slide 1.1-13
Example: Consider the function π π₯ =1
π₯
Find the following limits:
a) limπ₯β0β
π π₯
b) limπ₯β0+
π π₯
c) limπ₯β0
π π₯
graphically and numerically
1.1 Limits: A Numerical and Graphical Approach
2012 Pearson Education, Inc. All rights reserved Slide 1.1-14
Solutions:
a) limπ₯β0β
π π₯ = ββ
Means: As π₯ gets closer to 0 from
the left, π(π₯) gets more negative
b) limπ₯β0+
π π₯ = β
Means: As π₯ gets closer to 0 from
the right, π(π₯) gets more positive
c) limπ₯β0
π π₯ Does not exist
1.1 Limits: A Numerical and Graphical Approach
2012 Pearson Education, Inc. All rights reserved Slide 1.1-15
Important Point: β is not a number!
β’ A number is a destination on the number line
β’ β 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 π π₯ =1
π₯
Find the following limits:
a) limπ₯βββ
π π₯
b) limπ₯ββ
π π₯
graphically and numerically
1.1 Limits: A Numerical and Graphical Approach
2012 Pearson Education, Inc. All rights reserved Slide 1.1-17
Solutions:
a) limπ₯βββ
π π₯ = 0
Means: As π₯ gets more negative,
π(π₯) gets closer to 0
b) limπ₯ββ
π π₯ = 0
Means: As π₯ gets more positive,
π(π₯) 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
2012 Pearson Education, Inc. All rights reserved Slide 1.1-19
2012 Pearson Education, Inc. All rights reserved Slide 1.1-20
= 3.30, 3.30, 3.30
= 2.90, 3.30, Does not exist
2012 Pearson Education, Inc. All rights reserved Slide 1.1-21
Homework
β’ 1.1 HW
1.1 Limits: A Numerical and Graphical Approach