Aim: Do Limits at Infinity make sense?

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Aim Do Limits at Infinity make sense?
Do Now
List the characteristics of the following function
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Do Now
f(x) ?? 3 as x ? ?
f(x) ?? 3 as x ? -?
x -? ? -100 -10 -1 0 1 10 100 ? ?
f(x) 3 ? 2.9997 2.97 1.5 0 1.5 2.97 2.9997 ? 3
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Horizontal Asymptote
f(x) ?? 3 as x ? ?
f(x) ?? 3 as x ? -?
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Limits at Infinity for Rational Functions
For the rational function f(x) P(x)/Q(x),
where P(x) anxn . . . .a0 and Q(x) bmxm
. . . .b0 the limit as x approaches positive or
negative infinity is as follows If n gt m, the
limit does not exist.
n is the highest power of numerator m is highest
power of denominator
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Definition of Limits at Infinity
If f is a function and L1 and L2 are real
numbers, the statements
Limit as x approaches -?
Limit as x approaches ?
denote the limits at infinity. The first is read
the limit of f(x) as x approaches -? is L1 and
the second is read the limit of f(x) as x
approaches ? is L2.
If n is a positive real number, then
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Operations with Limits
If r is a positive rational number and c is any
real number, then Also, if xr is defined when x
lt 0, then
5 0 5
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Evaluating Limit at Infinity
Find the limit
Note a rational function will always approach
the same horizontal asymptote to the right and
left.
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Limits at Infinity
What happens to f(x) as x approaches ? infinity?
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Model Problem
Find the limit
The limit of f(x) as x approaches ? is 4.
Graphically
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Application
You are manufacturing a product that costs 0.50
per unit to produce. Your initial investment is
5000, which implies that the total cost of
producing x units is C 0.5x 5000. The
average cost per unit is Find the average cost
per unit when a) x 1000, b) x
10,000, c) when x ? ?
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Evaluating Limit at Infinity
Find the limit
algebraically
divide all terms by highest power of x in
denominator
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Model Problem
Find the limit for each as x approaches ?.
A polynomial tends to behave as its
highest-powered term behaves as x approaches ? ?
no limit
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Functions with Two Horizontal Asymptotes
Determine limits for each
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Functions with Two Horizontal Asymptotes
Determine limits for each
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Limits Involving Trig Functions
Determine limits for each
as x approaches infinity x oscillates between -1
and 1
conclusion
a limit does not exist
f(x) oscillates between two fixed values as x
approaches c.
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Limits Involving Trig Functions
Determine limits for each
Squeeze Theorem
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Model Problem 1
f(t) measures the level of oxygen in a pond,
where f(t) 1 is the normal (unpolluted) level
and the time t is measured in weeks. When t 0,
organic waste is dumped into the pond, and as the
waste material oxidizes, the level of oxygen in
the pond is What percent of the normal level of
oxygen exists in the pond after 1 week? After 2
weeks? After 10 weeks? What is the limit as t
approaches infinity?
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Model Problem 1
after 1 week? After 2 weeks? After 10
weeks? What is the limit as t approaches
infinity?
50
60
90.1
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Model Problem 1
What is the limit as t approaches infinity?
divide all terms by highest power of t in
denominator
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Limits at Infinity

• Let L be a real number.
• The statement
  means that for each ?? gt 0 there exists an M gt 0
  such that whenever x
  gt M.
• The statement
  means that for each ?? gt 0 there exists an N lt 0
  such that whenever x
  lt N.

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Limits at Infinity
For a given positive number ? there exists a
positive number M such that, for x gt M, the graph
of f will lie between the horizontal lines given
by y L ? and y L - ?.