applications of differentiation · 8/3/2015 · 3applications of differentiation. 2 limits at...
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3 Applications of Differentiation
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Limits at Infinity3.5
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• Determine (finite) limits at infinity.
• Determine the horizontal asymptotes, if any, of the graph of a function.
• Determine infinite limits at infinity.
Objectives
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Limits at InfinityHow do you determine end behavior for
polynomials?
How do you determine end behavior for rational functions?
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This section discusses the “end behavior” of a functionon an infinite interval. Consider the graph ofas shown in Figure 3.33.
Limits at Infinity
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Graphically, you can see that the values of f(x) appear to approach 3 as x increases without bound or decreases without bound. You can come to the same conclusions numerically, as shown in the table.
Limits at Infinity
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The table suggests that the value of f(x) approaches 3 as x increases without bound . Similarly, f(x) approaches 3 as x decreases without bound .These limits at infinity are denoted by
and
Limits at Infinity
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Horizontal Asymptotes
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How do we find horizontal asymptotes?
Horizontal Asymptotes
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Horizontal Asymptotes
(For #3: when the limit does not exist, the function goes to infinity or negative infinity as we go to the far left or right.)
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Example – Finding a Limit at Infinity
Find the limit:(Discuss Indeterminate Form: )
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So, the line y = 2 is a horizontal asymptote to the right.By taking the limit as , you can see that y = 2 is also a horizontal asymptote to the left.
Example – Finding a Limit at Infinity Find the limit:
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Example – Finding a Limit at Infinity
Find the limit:
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Example – Finding a Limit at Infinity Find the limit:
Solution: Using Theorem 3.10, you can write
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Example – Finding a Limit at Infinity
Find the limit:
Be careful!
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Example – Finding a Limit at Infinity
Find the limits:
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Example – Finding a Limit at Infinity
Find the limits:
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Infinite Limits at Infinity
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Find each limit.
Example – Finding Infinite Limits at Infinity
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Find each limit.
Solution:
As x increases without bound, x3 also increases without bound. So, you can write
As x decreases without bound, x3 also decreases without bound. So, you can write
Example – Finding Infinite Limits at Infinity
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Example – continued
The graph of f(x) = x3 in Figure 3.42 illustrates these two results. These results agree with the Leading Coefficient Test for polynomial functions.
cont’d
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Example
What if in the rational function, the numerator has the leading term with a power that is one greater than the leading term in the denominator? Does it have a limit as it goes to infinity or negative infinity?
(Do #16 from section 3.5 in the book together as a class.)
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Summary:
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Assignment:Section 3.5: problems 16, 1327 odd,
39, 5971 odd