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.
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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
2
Some limits do not exist
Worked example
limπ₯β 1
2π₯=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
3
definition of limit of a function
x
limπ₯βπ
π (π₯ )=πΏNotation
Symbolic meaning
English meaningFor all positive , there exists a positive such that confining x to between a β 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
y
L+
L-
a+a-4
definition of limit of a function
x
limπ₯βπ
π (π₯ )=πΏNotation
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 β and a and between a and a + , implies that f(x) is also confined to a vertical range between above and below y = L.
y
L+
L-
a+a- a
L
5
definition of limit of a function
x
limπ₯βπ
π (π₯ )=πΏNotation
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 β and a and between a and a + , implies that f(x) is also confined to a vertical range between above and below y = L.
y
L+
L-
a+a- a
L
6
definition of limit of a function
x
limπ₯βπ
π (π₯ )=πΏNotation
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 β and a and between a and a + , implies that f(x) is also confined to a vertical range between above and below y = L.
ππ
π
πΏπΏ
πΏ
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?
limπ₯βπ
π (π₯ )=πΏNotation
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 β and a and between a and a + , implies that f(x) is also confined to a vertical range between above and below y = L.
8
Some limits do not exist
Worked example
limπ₯β 1
2π₯=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
9
Definition of improper limit of a function
limπ₯βπ
π (π₯ )=+βNotation
Symbolic meaning
English meaningFor all positive , there exists a positive such that confining x to between a β 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
M
10
Definition of improper limit of a function
a+a-
limπ₯βπ
π (π₯ )=+βNotation
Symbolic meaning
English meaningFor all positive , there exists a positive such that confining x to between a β 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
11
Definition of improper limit of a function
a+a-
limπ₯βπ
π (π₯ )=+βNotation
Symbolic meaning
English meaningFor all positive , there exists a positive such that confining x to between a β 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
12
Definition of improper limit of a function
a+a-
limπ₯βπ
π (π₯ )=+βNotation
Symbolic meaning
English meaningFor all positive , there exists a positive such that confining x to between a β 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
π
πΏπΏ
πΏ
π π
M
13
Definition of improper limit of a function
a+a-
limπ₯βπ
π (π₯ )=+βNotation
Symbolic meaning
English meaningFor all positive , there exists a positive such that confining x to between a β 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
14
Some limits do not exist
Worked example
limπ₯β 1
2π₯=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
y
15
Definition of limit of a function as
x
limπ₯β+β
π (π₯ )=πΏNotation
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.
L+
L-
x
y
S16
Definition of limit of a function as
L
limπ₯β+β
π (π₯ )=πΏNotation
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.
L+
L-
x
y
S17
Definition of limit of a function as
L
limπ₯β+β
π (π₯ )=πΏNotation
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.
L+
L-
x
y
S18
Definition of limit of a function as
L
limπ₯β+β
π (π₯ )=πΏNotation
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. π
ππ
ππ
π
19
Some limits do not exist
Worked example
limπ₯β 1
2π₯=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
y
20
Improper limits at infinity
x
limπ₯β+β
π (π₯ )=+βNotation
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.
M
x
y
S21
Improper limits at infinity
limπ₯β+β
π (π₯ )=+βNotation
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.
M
x
y
S22
Improper limits at infinity
limπ₯β+β
π (π₯ )=+βNotation
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.
M
x
y
S23
Improper limits at infinity
limπ₯β+β
π (π₯ )=+βNotation
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.
ππ
π
ππ π
Worked example
limπ₯β 1
2π₯=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
yy
L+L-
L+
L-
a+a-25
Sometimes a limit does not exist (D.N.E.)
limπ₯β+β
π (π₯ )=D .N . E .Example: Limit as x becomes arbitrarily large
x
L
limπ₯βπ
π (π₯ )=D .N . E .Example: Limit of a function
a
L
xS
Worked example
limπ₯β 1
2π₯=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
y
2+
2-
1+1-
2
28
Example:
1x
π¦ππ΄π=2 (1+πΏ ) π¦ππΌπ=2 (1βπΏ )π¦ππ΄π=2+2Ξ΄ π¦ππΌπ=2β2πΏ
Pick an arbitrary
Pick an . Can we choose s.t. ?
|π¦ππ΄πβ2|<π |π¦ππΌπβ2|<π|(2+2πΏ )β2|<π |(2β2πΏ )β2|<π
|2 πΏ|<π |β2πΏ|<π2 πΏ<ππΏ<
π2
Works for any
yMAX
yMIN
limπ₯β 1
2π₯=2Show thatWant to show
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