# Limits at infinity - horizontal asymptotes

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##### Intros

##### Examples

###### Lessons

**Relate Asymptotes to Limits**

Express all asymptotes in limit notations for the function $f$ whose graph is shown below.

**Discuss the Foundation of Limits at Infinity**

Find:

i) $lim_{x \to \infty } \;\frac{1}{x}$

ii) $lim_{x \to - \infty } \;\frac{1}{x}$**Use "Highest Power Rule" to Evaluate Limits at Infinity of Rational Functions in 3 Types**

Find:- $lim_{x \to \infty } \;\frac{{-5x^2+13x+100}}{{2x^2-8}}$ (Type 1: degree of numerator = degree of denominator)
- $lim_{x \to - \infty } \;\frac{{2x - 9}}{{{x^3} + 7{x^2} + 10x + 21}}$ (Type 2: degree of numerator < degree of denominator)
- $lim_{x \to \infty } \;\frac{{{x^2} - 3x + 11}}{{5 - x}}$ (Type 3: degree of numerator > degree of denominator)

**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) $lim_{x \to - \infty } \;\frac{{\sqrt {3{x^2} + 7x - 1000} }}{{5x + 8}}$**Multiply Conjugates First, then Evaluate Limits**

Find:**Infinite Limits at Infinity**

Find:

i) $lim_{x \to \infty } \;{x^3}$

ii) $lim_{x \to - \infty } \;{x^3}$**Ambiguous Case:**$\infty - \infty$

Find $lim_{x \to \infty } \;{x^2} - x$**Limits at Infinity of Exponential Functions**

Find:

i) $lim_{x \to \infty } \;{e^x}$

ii) $lim_{x \to - \infty } \;{e^x}$**Limits at Infinity of Trigonometric Functions**

Find $lim_{x \to \infty } \;\sin x$