as x approaches a finite value
DESCRIPTION
Limits. f ( x ) can become arbitrarily large. f ( x ) can approach a finite value. As x approaches a finite value. As x becomes arbitrarily large. Limits. Show that. Some limits do not exist. f ( x ) can become arbitrarily large. Worked example. - PowerPoint PPT PresentationTRANSCRIPT
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1
As x approaches a finite value
f(x) can approach a finite value
f(x) can become arbitrarily large
As x becomes arbitrarily large
Limits
![Page 2: As x approaches a finite value](https://reader036.vdocuments.us/reader036/viewer/2022062520/56815abc550346895dc87be9/html5/thumbnails/2.jpg)
2
Some limits do not exist
Worked example
lim𝑥→ 1
2𝑥=2Show that
As x approaches a finite value
f(x) can approach a finite value
f(x) can become arbitrarily large
As x becomes arbitrarily large
Limits
![Page 3: As x approaches a finite value](https://reader036.vdocuments.us/reader036/viewer/2022062520/56815abc550346895dc87be9/html5/thumbnails/3.jpg)
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definition of limit of a function
x
lim𝑥→𝑎
𝑓 (𝑥 )=𝐿Notation
Symbolic meaning
English meaningFor all positive , there exists a positive such that confining x to between a – and a and between a and a + , implies that f(x) is also confined to a vertical range between above and below y = L.
VernacularAs x becomes arbitrarily close to a, f(x) becomes arbitrarily close to L.
y
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y
L+
L-
a+a-4
definition of limit of a function
x
lim𝑥→𝑎
𝑓 (𝑥 )=𝐿Notation
Symbolic meaning
VernacularAs x becomes arbitrarily close to a, f(x) becomes arbitrarily close to L.
a
L
English meaningFor all positive , there exists a positive such that confining x to between a – and a and between a and a + , implies that f(x) is also confined to a vertical range between above and below y = L.
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y
L+
L-
a+a- a
L
5
definition of limit of a function
x
lim𝑥→𝑎
𝑓 (𝑥 )=𝐿Notation
Symbolic meaning
VernacularAs x becomes arbitrarily close to a, f(x) becomes arbitrarily close to L.
English meaningFor all positive , there exists a positive such that confining x to between a – and a and between a and a + , implies that f(x) is also confined to a vertical range between above and below y = L.
![Page 6: As x approaches a finite value](https://reader036.vdocuments.us/reader036/viewer/2022062520/56815abc550346895dc87be9/html5/thumbnails/6.jpg)
y
L+
L-
a+a- a
L
6
definition of limit of a function
x
lim𝑥→𝑎
𝑓 (𝑥 )=𝐿Notation
Symbolic meaning
VernacularAs x becomes arbitrarily close to a, f(x) becomes arbitrarily close to L.
English meaningFor all positive , there exists a positive such that confining x to between a – and a and between a and a + , implies that f(x) is also confined to a vertical range between above and below y = L.
𝜖𝜖
𝜖
𝛿𝛿
𝛿
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y
L+
L-
a+a- a
L
7
definition of limit of a function
x
STOPCan L still be the limit of f(x) as x approaches a if, as illustrated, f(a) no longer equals L?
lim𝑥→𝑎
𝑓 (𝑥 )=𝐿Notation
VernacularAs x becomes arbitrarily close to a, f(x) becomes arbitrarily close to L.
English meaningFor all positive , there exists a positive such that confining x to between a – and a and between a and a + , implies that f(x) is also confined to a vertical range between above and below y = L.
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Some limits do not exist
Worked example
lim𝑥→ 1
2𝑥=2Show that
As x approaches a finite value
f(x) can approach a finite value
f(x) can become arbitrarily large
As x becomes arbitrarily large
Limits
![Page 9: As x approaches a finite value](https://reader036.vdocuments.us/reader036/viewer/2022062520/56815abc550346895dc87be9/html5/thumbnails/9.jpg)
9
Definition of improper limit of a function
lim𝑥→𝑎
𝑓 (𝑥 )=+∞Notation
Symbolic meaning
English meaningFor all positive , there exists a positive such that confining x to between a – and a and between a and a + , implies that f(x) exceeds .
VernacularAs x becomes arbitrarily close to a, f(x) becomes arbitrarily large.
x
y
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M
10
Definition of improper limit of a function
a+a-
lim𝑥→𝑎
𝑓 (𝑥 )=+∞Notation
Symbolic meaning
English meaningFor all positive , there exists a positive such that confining x to between a – and a and between a and a + , implies that f(x) exceeds .
VernacularAs x becomes arbitrarily close to a, f(x) becomes arbitrarily large.
ax
y
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M
11
Definition of improper limit of a function
a+a-
lim𝑥→𝑎
𝑓 (𝑥 )=+∞Notation
Symbolic meaning
English meaningFor all positive , there exists a positive such that confining x to between a – and a and between a and a + , implies that f(x) exceeds .
VernacularAs x becomes arbitrarily close to a, f(x) becomes arbitrarily large.
ax
y
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M
12
Definition of improper limit of a function
a+a-
lim𝑥→𝑎
𝑓 (𝑥 )=+∞Notation
Symbolic meaning
English meaningFor all positive , there exists a positive such that confining x to between a – and a and between a and a + , implies that f(x) exceeds .
VernacularAs x becomes arbitrarily close to a, f(x) becomes arbitrarily large.
ax
y
𝑀
𝛿𝛿
𝛿
𝑀 𝑀
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M
13
Definition of improper limit of a function
a+a-
lim𝑥→𝑎
𝑓 (𝑥 )=+∞Notation
Symbolic meaning
English meaningFor all positive , there exists a positive such that confining x to between a – and a and between a and a + , implies that f(x) exceeds .
VernacularAs x becomes arbitrarily close to a, f(x) becomes arbitrarily large.
ax
y
STOPWrite down the analogous definition for
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Some limits do not exist
Worked example
lim𝑥→ 1
2𝑥=2Show that
As x approaches a finite value
f(x) can approach a finite value
f(x) can become arbitrarily large
As x becomes arbitrarily large
Limits
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y
15
Definition of limit of a function as
x
lim𝑥→+∞
𝑓 (𝑥 )=𝐿Notation
Symbolic meaning
English meaningFor all positive , there exists a positive such that insisting that x exceed implies that f(x) is between above and below L.
VernacularAs x becomes arbitrarily large, f(x) becomes arbitrarily close to L.
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L+
L-
x
y
S16
Definition of limit of a function as
L
lim𝑥→+∞
𝑓 (𝑥 )=𝐿Notation
Symbolic meaning
English meaningFor all positive , there exists a positive such that insisting that x exceed implies that f(x) is between above and below L.
VernacularAs x becomes arbitrarily large, f(x) becomes arbitrarily close to L.
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L+
L-
x
y
S17
Definition of limit of a function as
L
lim𝑥→+∞
𝑓 (𝑥 )=𝐿Notation
Symbolic meaning
English meaningFor all positive , there exists a positive such that insisting that x exceed implies that f(x) is between above and below L.
VernacularAs x becomes arbitrarily large, f(x) becomes arbitrarily close to L.
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L+
L-
x
y
S18
Definition of limit of a function as
L
lim𝑥→+∞
𝑓 (𝑥 )=𝐿Notation
Symbolic meaning
English meaningFor all positive , there exists a positive such that insisting that x exceed implies that f(x) is between above and below L.
VernacularAs x becomes arbitrarily large, f(x) becomes arbitrarily close to L. 𝜖
𝜖𝜖
𝑆𝑆
𝑆
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19
Some limits do not exist
Worked example
lim𝑥→ 1
2𝑥=2Show that
As x approaches a finite value
f(x) can approach a finite value
f(x) can become arbitrarily large
As x becomes arbitrarily large
Limits
![Page 20: As x approaches a finite value](https://reader036.vdocuments.us/reader036/viewer/2022062520/56815abc550346895dc87be9/html5/thumbnails/20.jpg)
y
20
Improper limits at infinity
x
lim𝑥→+∞
𝑓 (𝑥 )=+∞Notation
Symbolic meaning
English meaningFor all positive , there exists a positive such that insisting that x exceed implies that f(x) exceeds .
VernacularAs x becomes arbitrarily large, f(x) becomes arbitrarily large.
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M
x
y
S21
Improper limits at infinity
lim𝑥→+∞
𝑓 (𝑥 )=+∞Notation
Symbolic meaning
English meaningFor all positive , there exists a positive such that insisting that x exceed implies that f(x) exceeds .
VernacularAs x becomes arbitrarily large, f(x) becomes arbitrarily large.
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M
x
y
S22
Improper limits at infinity
lim𝑥→+∞
𝑓 (𝑥 )=+∞Notation
Symbolic meaning
English meaningFor all positive , there exists a positive such that insisting that x exceed implies that f(x) exceeds .
VernacularAs x becomes arbitrarily large, f(x) becomes arbitrarily large.
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M
x
y
S23
Improper limits at infinity
lim𝑥→+∞
𝑓 (𝑥 )=+∞Notation
Symbolic meaning
English meaningFor all positive , there exists a positive such that insisting that x exceed implies that f(x) exceeds .
VernacularAs x becomes arbitrarily large, f(x) becomes arbitrarily large.
𝑆𝑆
𝑆
𝑀𝑀 𝑀
![Page 24: As x approaches a finite value](https://reader036.vdocuments.us/reader036/viewer/2022062520/56815abc550346895dc87be9/html5/thumbnails/24.jpg)
Worked example
lim𝑥→ 1
2𝑥=2Show that
24
Some limits do not exist
As x approaches a finite value
f(x) can approach a finite value
f(x) can become arbitrarily large
As x becomes arbitrarily large
Limits
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yy
L+L-
L+
L-
a+a-25
Sometimes a limit does not exist (D.N.E.)
lim𝑥→+∞
𝑓 (𝑥 )=D .N . E .Example: Limit as x becomes arbitrarily large
x
L
lim𝑥→𝑎
𝑓 (𝑥 )=D .N . E .Example: Limit of a function
a
L
xS
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Worked example
lim𝑥→ 1
2𝑥=2Show that
26
Some limits do not exist
As x approaches a finite value
f(x) can approach a finite value
f(x) can become arbitrarily large
As x becomes arbitrarily large
Limits
![Page 27: As x approaches a finite value](https://reader036.vdocuments.us/reader036/viewer/2022062520/56815abc550346895dc87be9/html5/thumbnails/27.jpg)
y
2+
2-
2
1+1-27
Example:
lim𝑥→ 1
2𝑥=2Show that
1x
Want to show
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y
2+
2-
1+1-
2
28
Example:
1x
𝑦𝑀𝐴𝑋=2 (1+𝛿 ) 𝑦𝑀𝐼𝑁=2 (1−𝛿 )𝑦𝑀𝐴𝑋=2+2δ 𝑦𝑀𝐼𝑁=2−2𝛿
Pick an arbitrary
Pick an . Can we choose s.t. ?
|𝑦𝑀𝐴𝑋−2|<𝜖 |𝑦𝑀𝐼𝑁−2|<𝜖|(2+2𝛿 )−2|<𝜖 |(2−2𝛿 )−2|<𝜖
|2 𝛿|<𝜖 |−2𝛿|<𝜖2 𝛿<𝜖𝛿<
𝜖2
Works for any
yMAX
yMIN
lim𝑥→ 1
2𝑥=2Show thatWant to show
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29
Some limits do not exist
Worked example
lim𝑥→ 1
2𝑥=2Show that
As x approaches a finite value
f(x) can approach a finite value
f(x) can become arbitrarily large
As x becomes arbitrarily large
Limits