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 xx

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-3x 2 12.

1-x 13.

53x 44.

3x5.

16x 126. 3 5 2

xx x

x

x x

xx

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. limx

1 1 12. a. lim b. lim c. limx x xx-3 x-3 x-33. 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. lim3

2. lim1

3. lim7

4. lim49

25. lim4 12

6. lim tan

17. lim 1

x

x

x

x

x

x

x

xxxxx

x

xxx

x xx 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. lim3 52 75. lim3 5

6. lim sin

x

x

x

x

x

x

xx

xxxx xxx xxx

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. lim3 13 12. lim4 5

3. lim3

1 64. lim1 52 65. lim

1

1 56. lim3 2

x

x

x

x

x

x

xxxxxx

xx

xx

xx x

Theorem

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

ax

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. Rationalizing3. Rewriting the problem

Examples: Divide

2

2

2

4

2

3 11. lim2 3

22. lim2

3 23. lim2 1

3 24. lim

x

x

x

x

xxx

x xxx

x xx

Example: Rationalize

2

2

11. lim1

2. lim 1

3. lim 4

4. lim 1

x

x

x

x

xx

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

1cos4.lim

2 115. lim 3

26.lim1

7.lim 3 1

x

x

x

x

x

x

x

x xxx x

x x

xx

x

xx

x xx x x

x

End Behavior Models

• Graphon 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 xx xx x xf xx x x

HW: p. 71

• 1-22,29-38• Worksheet