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When you are done with your homework, you should be able to… Determine (finite) limits at infinity Determine the horizontal asymptotes, if any, of the graph of a function Determine infinite limits at infinity
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LIMITS AT INFINITY
Section 3.5Section 3.5
When you are done with your homework, you should
be able to…
• Determine (finite) limits at infinity• Determine the horizontal
asymptotes, if any, of the graph of a function
• Determine infinite limits at infinity
Pythagoras lived in 550BC. He was the 1st person to teach that nature is governed by mathematics. Another
of his achievements was:A. The mathematics of musical harmony—2
tones harmonize when the ratio of their frequencies form a “simple fraction”.
B. He taught that atoms were in the shape of regular polyhedra.
C. He started a new religion (secret society) which greatly influenced western religion.
D. A and C.
DEFINITION OF LIMITS AT INFINITY
Let L be a real number.• The statement means that
for each there exists an such that whenever
• The statement means that for each there exists an such that whenever
limx
f x L
O
f x L M O
.x M
limx
f x L
N O
f x L O
.x N
THEOREM: LIMITS AT INFINITY
If r is a positive rational number and c is any real number, then
If is defined when then •
lim 0.rx
cx
rx 0,x
lim 0.rx
cx
Evaluate the following limit.
0.00.0
3
5limx x
HORIZONTAL ASYMPTOTES
The line is a horizontal asymptote of the graph of f if
y L
lim or lim .x x
f x L f x L
GUIDELINES FOR FINDING LIMITS AT OF RATIONAL FUNCTIONS
1. If the degree of the numerator is less than the degree of the denominator, then the limit of the rational function is 0.
2. If the degree of the numerator is equal to the degree of the denominator, then the limit of the rational function is the ratio of the leading coefficients.
3. If the degree of the numerator is greater than the degree of the denominator, then the finite limit of the rational function does not exist.
Find the horizontal asymptote(s) of the following function.
A. 3, -4B. 2C. y = 2D. B and CE. None of the above.
2
2
2 15 612
x xf xx x
Find the horizontal asymptote(s) of the following function.
A.
B.
C. D. No horizontal asymptotes.
5
5
4 5 76 1x xg xx
46
y
23
y
23
y
DEFINITION OF INFINITE LIMITS AT INFINITY
Let f be a function defined on the interval 1.The statement means that for each
positive number M there is a corresponding number such that
whenever2.The statement means that for
each negative number M there is a corresponding number such that
whenever
limx
f x
N O.x N
limx
f x
f x M
0N
, .a
f x M .x N
Find the horizontal asymptote(s) of the following function.
A.
B.
C. D. No horizontal asymptotes.
4 3
3
3 2 5 72 7
x x xr sx
32
y
y
32
y
52 6lim2x
x xx
