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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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Presentation on theme: "3.5 Limits at Infinity  Determine limits at infinity  Determine the horizontal asymptotes, if any, of the graph of function. Standard 4.5a."— Presentation transcript:

1 3.5 Limits at Infinity  Determine limits at infinity  Determine the horizontal asymptotes, if any, of the graph of function. Standard 4.5a

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

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

4 Limit at negative infinity Limit at positive infinity

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

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

8 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

9 Finding Limits at Infinity

10 is an indeterminate form

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

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14 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.

15 For x < 0, you can write

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

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


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