1 as x approaches a finite value f(x) can approach a finite value f(x) can become arbitrarily large...
TRANSCRIPT
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
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
3
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
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.
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.
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
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.
8
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
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
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
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
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
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
14
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
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.
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.
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.
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
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
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.
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.
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.
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
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
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
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
y
2+
2-
2
1+1-27
Example:
lim𝑥→ 1
2𝑥=2Show that
1x
Want to show
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
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