2.2 Limits Involving Infinity
Jan 02, 2016
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