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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 -∞ -100 -10 -1 1 10 100 ∞ f(x)

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**x decreases x increases f(x) approaches 2 f(x) approaches 2 x -∞ -100**

1 10 100 ∞ f(x) 2 1.99 1.96 .667 f(x) approaches 2 f(x) approaches 2

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**Limit at negative infinity**

Limit at positive infinity

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**To Infinity and Beyond…**

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 xr is defined when x < 0, then

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**Finding Limits at Infinity**

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**Finding Limits at Infinity**

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**

If the degree of the numerator is < the degree of the denominator, then the limit is 0. If the degree of the numerator = the degree of the denominator, then the limit is the ratio of the leading coefficients. 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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