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.

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