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Limits

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Limits

y

y

y

Limit of a function

Improper limits

Limits “at” infinity

Show that

y

Worked example

Sometimes limits do not exist

x

x

x

x

definition of limit of a function

y

Notation

Vernacular

As x becomes arbitrarily close to p, f(x) becomes arbitrarily close to L.

Symbolic meaning

English meaning

For all positive , there exists a positive such that confining x to between p – and p and between p and p + , implies that f(x) is also confined to a vertical range between above and below y = L.

x

definition of limit of a function

y

Notation

L+

Vernacular

As x becomes arbitrarily close to p, f(x) becomes arbitrarily close to L.

L

Symbolic meaning

English meaning

For all positive , there exists a positive such that confining x to between p – and p and between p and p + , implies that f(x) is also confined to a vertical range between above and below y = L.

L-

x

p+

p-

p

definition of limit of a function

y

Notation

Vernacular

As x becomes arbitrarily close to p, f(x) becomes arbitrarily close to L.

L+

L

Symbolic meaning

L-

English meaning

For all positive , there exists a positive such that confining x to between p – and p and between p and p + , implies that f(x) is also confined to a vertical range between above and below y = L.

x

p+

p-

p

definition of limit of a function

y

Notation

Vernacular

As x becomes arbitrarily close to p, f(x) becomes arbitrarily close to L.

L+

L

Symbolic meaning

L-

English meaning

x

p+

p-

p

definition of limit of a function

y

Notation

Vernacular

As x becomes arbitrarily close to p, f(x) becomes arbitrarily close to L.

L+

Can L still be the limit of f(x) as x approaches p if, as illustrated, f(p) no longer equals L?

L

L-

English meaning

STOP

x

p+

p-

p

Limits

y

y

y

Limit of a function

Improper limits

Limits “at” infinity

Show that

y

Worked example

Sometimes limits do not exist

x

x

x

x

Definition of improper limit of a function

y

Notation

Vernacular

As x becomes arbitrarily close to p, f(x) becomes arbitrarily large.

Symbolic meaning

English meaning

For all positive , there exists a positive such that confining x to between p – and p and between p and p + , implies that f(x) exceeds .

x

Definition of improper limit of a function

y

Notation

Vernacular

As x becomes arbitrarily close to p, f(x) becomes arbitrarily large.

M

Symbolic meaning

English meaning

For all positive , there exists a positive such that confining x to between p – and p and between p and p + , implies that f(x) exceeds .

x

p+

p-

p

Definition of improper limit of a function

y

Notation

Vernacular

As x becomes arbitrarily close to p, f(x) becomes arbitrarily large.

M

Symbolic meaning

English meaning

For all positive , there exists a positive such that confining x to between p – and p and between p and p + , implies that f(x) exceeds .

x

p+

p-

p

Definition of improper limit of a function

y

Notation

Vernacular

As x becomes arbitrarily close to p, f(x) becomes arbitrarily large.

M

Symbolic meaning

English meaning

x

p+

p-

p

Definition of improper limit of a function

y

Notation

Vernacular

As x becomes arbitrarily close to p, f(x) becomes arbitrarily large.

M

Symbolic meaning

English meaning

Write down the analogous definition for

STOP

x

p+

p-

p

Limits

y

y

y

Limit of a function

Improper limits

Limits “at” infinity

Show that

y

Worked example

Sometimes limits do not exist

x

x

x

x

Definition of limit of a function as

y

Notation

Vernacular

As x becomes arbitrarily large, f(x) becomes arbitrarily close to L.

Symbolic meaning

English meaning

For all positive , there exists a positive such that insisting that x exceed implies that f(x) is between above or below L.

x

Definition of limit of a function as

y

Notation

L+

Vernacular

As x becomes arbitrarily large, f(x) becomes arbitrarily close to L.

L

Symbolic meaning

L-

English meaning

For all positive , there exists a positive such that insisting that x exceed implies that f(x) is between above or below L.

x

S

Definition of limit of a function as

y

Notation

Vernacular

As x becomes arbitrarily large, f(x) becomes arbitrarily close to L.

L+

L

L-

Symbolic meaning

English meaning

For all positive , there exists a positive such that insisting that x exceed implies that f(x) is between above or below L.

x

S

Definition of limit of a function as

y

Notation

Vernacular

As x becomes arbitrarily large, f(x) becomes arbitrarily close to L.

L+

L

L-

Symbolic meaning

English meaning

x

S

Limits

y

y

y

Limit of a function

Improper limits

Limits “at” infinity

Show that

y

Worked example

Sometimes limits do not exist

x

x

x

x

Sometimes a limit does not exist (D.N.E.)

Example: Limit as x becomes arbitrarily large

Example: Limit of a function

y

y

L+

L+

L

L

L-

L-

x

x

p+

p-

S

p

Limits

y

y

y

Limit of a function

Improper limits

Limits “at” infinity

Show that

y

Worked example

Sometimes limits do not exist

x

x

x

x

Example:

Want to show

Show that

y

2+

2

2-

x

1+

1-

1

Example:

Want to show

Works for any

Show that

Pick an arbitrary

y

2+

Pick an . Can we choose s.t.?

2

yMAX

2-

yMIN

x

1+

1-

1