# Limits at Infinity and Horizontal Asymptotes Explained Dive into the world of limits at infinity and horizontal asymptotes. Master essential calculus concepts, analyze function behavior, and apply your knowledge to real-world scenarios.

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Intros

Examples

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