# Limits at infinity - horizontal asymptotes

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##### Intros
###### Lessons
1. Introduction to Horizontal Asymptotes
2. opposite relationship between "vertical asymptote" and "horizontal asymptote"
3. how horizontal asymptotes are defined on each end of a function
4. evaluate limits at infinity algebraically –"Highest Power Rule"!
5. lesson overview
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##### Examples
###### Lessons
1. Relate Asymptotes to Limits
Express all asymptotes in limit notations for the function $f$ whose graph is shown below.

1. Discuss the Foundation of Limits at Infinity
Find:
i) $lim_{x \to \infty } \;\frac{1}{x}$
ii) $lim_{x \to - \infty } \;\frac{1}{x}$
1. Use "Highest Power Rule" to Evaluate Limits at Infinity of Rational Functions in 3 Types
Find:
1. $lim_{x \to \infty } \;\frac{{-5x^2+13x+100}}{{2x^2-8}}$
(Type 1: degree of numerator = degree of denominator)
2. $lim_{x \to - \infty } \;\frac{{2x - 9}}{{{x^3} + 7{x^2} + 10x + 21}}$
(Type 2: degree of numerator < degree of denominator)
3. $lim_{x \to \infty } \;\frac{{{x^2} - 3x + 11}}{{5 - x}}$
(Type 3: degree of numerator > degree of denominator)
2. Evaluate Limits at Infinity of Functions Involving Radicals
Find the horizontal asymptotes of the function $f\left( x \right) = \frac{{\sqrt {3{x^2} + 7x - 1000} }}{{5x + 8}}$ by evaluating:
i) $lim_{x \to \infty } \;\frac{{\sqrt {3{x^2} + 7x - 1000} }}{{5x + 8}}$
ii) $lim_{x \to - \infty } \;\frac{{\sqrt {3{x^2} + 7x - 1000} }}{{5x + 8}}$
1. Multiply Conjugates First, then Evaluate Limits
Find:
1. $lim_{x \to \infty } \;\left( {\sqrt {9{x^2} + 12x} - 3x} \right)$
2. $lim_{x \to - \infty } \;\left( {x + \sqrt {{x^2} - 5x} } \right)$
2. Infinite Limits at Infinity
Find:
i) $lim_{x \to \infty } \;{x^3}$
ii) $lim_{x \to - \infty } \;{x^3}$
1. Ambiguous Case: $\infty - \infty$
Find $lim_{x \to \infty } \;{x^2} - x$
1. Limits at Infinity of Exponential Functions
Find:
i) $lim_{x \to \infty } \;{e^x}$
ii) $lim_{x \to - \infty } \;{e^x}$
1. Limits at Infinity of Trigonometric Functions
Find $lim_{x \to \infty } \;\sin x$