SLIDE 1
Limits
Michael Freeze
MAT 151 UNC Wilmington
Summer 2013
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Section 3.1 :: Limits
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Limits Michael Freeze MAT 151 UNC Wilmington Summer 2013 1 / 18 - - PowerPoint PPT Presentation
Limits Michael Freeze MAT 151 UNC Wilmington Summer 2013 1 / 18 - - PowerPoint PPT Presentation
Limits Michael Freeze MAT 151 UNC Wilmington Summer 2013 1 / 18 Section 3.1 :: Limits 2 / 18 Informal Definition of Limit We write x a f ( x ) = L lim and say the limit of f ( x ), as x approaches a , equals L if we can make
SLIDE 2
SLIDE 3
Informal Definition of Limit
We write lim
x→a f (x) = L
and say “the limit of f (x), as x approaches a, equals L” if we can make the values of f (x) arbitrarily close to L (as close to L as we like) by taking x sufficiently close to a (on either side of a), but not equal to a.
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SLIDE 4
Find the Value of the Limit lim
x→2
- x2 − 3x + 1
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SLIDE 5
Find the Value of the Limit lim
x→3
x2 − x − 1 √ x + 1
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SLIDE 6
Find the Value of the Limit lim
x→−2
x2 − x − 6 x + 2
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SLIDE 7
Find the Value of the Limit lim
x→25
√x − 5 x − 25
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SLIDE 8
One-Sided Limits
We write
lim
x→a− f (x) = L and say the left-hand limit of f (x) as x approaches a [or the limit of f (x) as x approaches a from the left] is equal to L if we can make the values of f (x) arbitrarily close to L by taking x to be sufficiently close to a with x less than a.
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SLIDE 9
One-Sided Limits
We write
lim
x→a+ f (x) = L and say the right-hand limit of f (x) as x approaches a [or the limit of f (x) as x approaches a from the right] is equal to L if we can make the values of f (x) arbitrarily close to L by taking x to be sufficiently close to a with x greater than a.
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SLIDE 10
Splitting Limits by Sides lim
x→a f (x) = L
if and only if lim
x→a− f (x) = L and
lim
x→a+ f (x) = L
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SLIDE 11
Informal Definition of Limit at Infinity
Let f be a function defined on some interval (a, ∞). Then lim
x→∞ f (x) = L
means that the values of f (x) can be made arbitrarily close to L by taking x sufficiently large.
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SLIDE 12
Horizontal Asymptotes
The line y = L is called a horizontal asymptote of the curve y = f (x) if either lim
x→∞ f (x) = L
- r
lim
x→−∞ f (x) = L
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SLIDE 13
Determine the Limit at Infinity lim
x→∞
5x − 1 2x + 2
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SLIDE 14
Determine the Limit at Infinity lim
x→∞
15x2 − 7x − 1 3x2 + 100
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SLIDE 15
Determine the Limit at Infinity lim
x→∞
x − 1 3x2 + 2
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SLIDE 16
Determine the Limit at Infinity lim
x→∞
x2 − x 5x + 4
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SLIDE 17
Average Cost
The cost (in dollars) for manufacturing a particular DVD is C(x) = 15, 000 + 6x, where x is the number of DVDs produced. Recall that the average cost per DVD, denoted by C(x), is found by dividing C(x) by x. Find and interpret lim
x→∞ C(x).
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SLIDE 18
Sediment
To develop strategies to manage water quality in polluted lakes, biologists must determine the depths
- f sediments and the rate of sedimentation. It has
been determined that the depth of sediment D(t) (in centimeters) with respect to time t (in years before 1990) for Lake Coeur d’Alene, Idaho, can be estimated by the equation D(t) = 155(1 − e−0.0133t), Source: Mathematics Teacher. (a) Find D(20) and interpret. (b) Find lim
t→∞ D(t) and interpret.
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