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1.5: Limits Involving Infinity Learning Goals: © 2009 Mark Pickering Calculate limits as Identify vertical and horizontal asymptotes

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Example Find the limit if it exists: In the previous lesson, we had this problem:

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Example Find the limit if it exists: How does this problem differ from the previous problem?

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Important Idea The above symbols describe the increasing or decreasing of a value without bound. Infinity is not a number.

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Example Use the table feature of your calculator to estimate: if it exists.

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Definition if they exist, means y = b is a horizontal asymptote b or

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Try This Use the graph and table features of your calculator to find any horizontal asymptotes for:

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Try This Use the graph and table features of your calculator to find any horizontal asymptotes for: &

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Important Idea Functions involving radicals may have 2 horizontal asymptotes Horizontal asymptotes are always written as

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Example Find the limit, if it exists: Indeter- minate form Divide top & bottom by highest power of x in denominator.

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Try This Find the limit, if it exists: DNE or

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Try This Find the limit, if it exists: 0

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Try This Find the limit, if it exists:

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Analysis In the last 3 examples, do you see a pattern? The highest power term is most influential.

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Important Idea If the degree of the top is greater than the degree of the bottom, For any rational function

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Important Idea If the degree of the bottom is greater than the degree of the top, For any rational function

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Important Idea If the degree of the top is the same as the degree of the top, the limit as For any rational function is the ratio of the leading coefficients.

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Try This Find the limit if it exists:

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Example Functions may approach different asymptotes as and as Consider each limit separately…

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Example As eventually x >0. Divide radical by and divide non-radical by x. Find the limit:

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Example As eventually x <0. Divide radical by and divide non-radical by x. Find the limit:

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Try This Using algebraic techniques, find the limit if it exists. Confirm your answer with your calculator.

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Solution

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Analysis The sine function oscillates between +1 and -1 1

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Analysis 1 The limit does not exist due to oscillation.

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Analysis Consider

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Analysis and Therefore, by the Sandwich Theorem,

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A function has an infinite limit at a if f(a) as x a. f(x) is unbounded at x=a. f(x) Definition

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A function has an infinite limit at a if f(a) as x a. f(x) is unbounded at x=a. f(x) Definition The line x = a is a vertical asymptote

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Important Idea A vertical asymptote is written x = a A horizontal asymptote is written y = a where a is any real number.

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Try This What is the limit as x 1 from the left and from the right? x =1 What is the vertical asymptote?

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Solution x =1 Vertical asymptote: x =1

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Try This What is the limit as x 1 from the left and from the right? x =1 What is the vert. asymptote?

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Solution x =1 Vertical asymptote: x =1

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Definition The value(s) that make the denominator of a rational function zero is a vertical asymptote.

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Example Determine all vertical asymptotes of Steps: 1. Factor & cancel if possible 2. Set denominator to 0 & solve

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Example Find the limit if it exists: 1. When you substitute x =1, do you get a number /0 or 0/0? The questions… 2. What is happening at x = a value larger than 1?

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Try This Find the limit if it exists: +

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Find all vertical asymptotes for for Hint: since, Example where does ?

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Lesson Close Limits are the foundation of both differential and integral Calculus. We will develop these ideas in chapter 3.

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Practice 76/1-7,13,15,27-33,59 (all odd)

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Limits at Infinity: End behavior of a Function

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