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3.5 Limits at Infinity Determine limits at infinity Determine the horizontal asymptotes, if any, of the graph of function. Standard 4.5a

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Do Now: Complete the table. x-∞ ∞ f(x)

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x-∞ ∞ f(x) x decreasesx increasesf(x) approaches 2

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Limit at negative infinity Limit at positive infinity

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We want to investigate what happens when functions go To Infinity and Beyond…

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Definition of a Horizontal Asymptote The line y = L is a horizontal asymptote of the graph of f if

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Limits at Infinity If r is a positive rational number and c is any real number, then Furthermore, if x r is defined when x < 0, then

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Finding Limits at Infinity

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is an indeterminate form

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Divide numerator and denominator by highest degree of x Simplify Take limits of numerator and denominator

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Guidelines for Finding Limits at ± ∞ of Rational Functions 1.If the degree of the numerator is < the degree of the denominator, then the limit is 0. 2.If the degree of the numerator = the degree of the denominator, then the limit is the ratio of the leading coefficients. 3.If the degree of the numerator is > the degree of the denominator, then the limit does not exist.

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For x < 0, you can write

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Limits Involving Trig Functions As x approaches ∞, sin x oscillates between -1 and 1. The limit does not exist. By the Squeeze Theorem

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Sketch the graph of the equation using extrema, intercepts, and asymptotes.

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