2.2 Limits Involving Infinity

The symbol

• The symbol means unbounded in the positive direction. (- in negative direction)

• It is NOT a number!

1Consider f x

x

As the denominator gets larger, the value of the fraction gets smaller. In other words as x gets larger positively or negatively, the y-values get closer to zero.

The line y = b is a horizontal asymptote if:

limx

f x b

or limx

f x b

The line y = 0 is a horizontal asymptote for f

0

1limx x

As the denominator approaches zero from the left, the value of the fraction gets very large.

0

1limx x

vertical asymptote at x=0.

As the denominator approaches zero from the right, the value of the fraction gets very large negatively.

Review: Finding Asymptotes• 1st make sure R(x) = p(x)/q(x) is in simplest

termsVertical Horizontal Oblique

Deg top > deg bottom

Set bottom = to 0 and solve

for x

none Divide top by bottom,

y=answer (no remainder)

Deg top = deg bottom

Set bottom = to 0 and solve

for x

y = quotient of leading

coeff of top and bottom

none

Deg top < deg bottom

Set bottom = to 0 and solve

for x

y = 0 none

Examples: Find asymptotes and graph

4 2

2

2

4

3

3

4

2

2

4x1.

x-3

x 2 12.

1

-x 13.

5

3x 44.

3

x5.

1

6x 126.

3 5 2

x

x x

x

x x

x

x

x x

Vertical Asymptotes- Infinite Limits

• The vertical line x = a is a vertical asymptote of a function y = f(x) if

• If

• If

lim or limx a x a

f x f x

x a

lim lim , then limx a x a

f x f x f x

x a

lim lim , then limx a x a

f x f x f x

Graphically

lim

x af x lim

x af x

lim does not existx af x

Examples: Find the limits graphically and numerically

00 0

2 2 200 0

2 2 211 1

1 1 11. a. lim b. lim c. lim

x1 1 1

2. a. lim b. lim c. limx x xx-3 x-3 x-3

3. a. lim b. lim c. lim x 1 x 1 x 1

xx x

xx x

xx x

x x

Examples: Find the limits graphically and numerically

3

1

27

27

26

2

2

0

41. lim

3

2. lim1

3. lim7

4. lim49

25. lim

4 12

6. lim tan

17. lim 1

x

x

x

x

x

x

x

x

xx

xx

x

x

xx

x x

x x

x

Horizontal Asymptotes – Limits at Infinity

• The line y = b is a horizontal asymptote of

y = f(x) if either

The limit at infinity is also referred to as end behavior.

lim or limx x

f x b f x b

Examples: Find the limits at infinity graphically and numerically

2

2

2

2

2

11. lim 2

2. lim1

sin3. lim

2 74. lim

3 5

2 75. lim

3 56. lim sin

x

x

x

x

x

x

x

x

xx

x

x x

x

x x

xx

Finding the limit at infinity analytically

• If f(x) is a rational function then to find the limit at infinity simply find the horizontal asymptote using the rules about degrees.

Examples

2

2

2

2

2

2

2

2 31. lim

3 1

3 12. lim

4 5

3. lim3

1 64. lim

1 5

2 65. lim

1

1 56. lim

3 2

x

x

x

x

x

x

x

x

x

x

x

xx

x

x

x

x

x x

Theorem

lim 0 where a is any real number and n>0nx

a

x

Non-rational functions

• If the function is not a rational function then you can try:

1. Dividing top and bottom by highest power on bottom

2. Rationalizing

3. Rewriting the problem

Examples: Divide

2

2

2

4

2

3 11. lim

2 32

2. lim2

3 23. lim

2 1

3 24. lim

x

x

x

x

x

xx

x xx

x

x x

x

Example: Rationalize

2

2

11. lim

1

2. lim 1

3. lim 4

4. lim 1

x

x

x

x

x

x

x x

x x

x x

Example: Rewrite

2 2

2

2

2

2

2 3

12

5 sin1.lim

2.lim sin cos

3.lim ln 2 ln 1

1cos

4.lim2 1

15. lim 3

26.lim

1

7.lim 3 1

x

x

x

x

x

x

x

x x

x

x x

x x

xx

x

xx

x x

x x x

x

End Behavior Models

• Graph

on the window

[-20, 20] by [-1000000, 5000000]

Notice as the graphs become identical.

We say that g(x) act as a model for f(x) as

or g(x) is an end behavior model for f(x)

4 3 2 43 2 3 5 6 and g x 3f x x x x x x

and x x

and x x

Example

• Show graphically that g(x) = x is a right end behavior model and h(x) = e-x is a left end behavior model for f(x) = x + e-x

End behavior models for polynomials

• If

1 2 21 2 2 1 0 ...

then g x is an end behavior model of f(x).

n n nn n n

nn

f x a x a x a x a x a x a

a x

Examples: Find the end behavior model

5 4 2

2

3 2

3 2

2 11.

3 5 7

2 12.

5 5

x x xf x

x x

x x xf x

x x x

HW: p. 71

• 1-22,29-38

• Worksheet