AP Calculus BCTuesday, 08 September 2015

• OBJECTIVE TSW (1) study and use a formal definition of a limit.

• WS Practice with Limits is due Friday, 09/11/15.

• T-shirt money due today!!!

• Test on Chapter 2 is Friday.

Formal Limits and Limits at Infinity

A Formal Definition of Limit- Definition

Let f be a function defined on an open interval containing c (except possibly at c) and let L be a . The statement

means that for each > 0, a > 0 if

lim

x cf x L

0 , then .x c f x L

A real numberEpsilon

“There exists”Delta

“Such That”

A Formal Definition of Limit

- Definition

Formal Limits and Limits at Infinity

Formal Limits and Limits at Infinity

A Formal Definition of Limit

Ex: Given that

Find such that

whenever

2

lim 5 4 6.x

x

5 4 6 0.01x

0 2 .x

5 4 6 0.01x

5 10 0.01x

5 2 0.01x

2 0.002x 0.002

Formal Limits and Limits at Infinity

A Formal Definition of Limit

Ex: Given that

Find such that

whenever

3

lim 3 2 11.x

x

3 2 11 0.01x

0 3 .x

3 2 11 0.01x

3 9 0.01x

3 3 0.01x

3 0.003x 0.003

Formal Limits and Limits at Infinity

Formal Limits and Limits at Infinity

Formal Limits and Limits at Infinity

Limits at infinity have many of the same properties of limits at specific values.

For example, if

lim and lim both exist, thenx x

f x g x

a lim lim lim)x x x

f x g x f x g x

b lim lim l) imx x x

f x g x f x g x

Similar properties hold for .

Formal Limits and Limits at Infinity

Formal Limits and Limits at Infinity

Ex: Find each limit:

2

3 5lim

5a)

2x

x

x

0

2 2

2

2 2

3 5

lim5 2x

xx xxx x

2

2

3 5

lim2

5x

x x

x

0 0

5 0

Divide numerator and denominator by the highest power of the variable.

2

2

1

1x

x

Apply “Limits at Infinity” theorem.

2

2

3 5lb i) m

5 2x

x

x

Formal Limits and Limits at Infinity

Ex: Find each limit:

3

5

2

2 2

2

2 2

3 5

lim5 2x

xx xxx x

2

2

53

lim2

5x

x

x

3 0

5 0

Ex: Find each limit:

Formal Limits and Limits at Infinity

3

2

3 5lc i) m

5 2x

x

x

DNE

3

3 3

2

3 3

3 5

lim5 2x

xx xxx x

3

3

53

lim5 2x

x

x x

3 0

0 0

Formal Limits and Limits at Infinity

How to find horizontal asymptotes:

1) Degree of Num. < Degree of Denom.HA: y = 0

2) Degree of Num. = Degree of Denom.HA:

3) Degree of Num. > Degree of Denom.HA: none

leading coefficient num.

leading coefficient denom.y

Formal Limits and Limits at Infinity

Ex: Find each limit:

2

5 3lim

7 2a)

x

x

x

2

5 3

lim7 2x

xx x

xx

2

2

35

lim7 2x

x

x

x

2

2 2

35

lim7 2x

x

x

x x

2

35

lim2

7x

x

x

5 0

7 0

5

7

Formal Limits and Limits at Infinity

Ex: Find each limit:

2

3 2lim

5 9b)

x

x

x

2

3 2

lim5 9x

xx x

xx

2

2

23

lim5 9x

x

x

x

2

2 2

23

lim5 9x

x

x

x x

2

23

lim9

5x

x

x

3 0

5 0

3

5

2

NOTE: If 0,x

x x