# Limits at infinity - horizontal asymptotes

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##### Intros

###### Lessons

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

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###### Topic Notes

There are times when we want to see how a function behaves near a horizontal asymptote. Much like finding the limit of a function as x approaches a value, we can find the limit of a function as x approaches positive or negative infinity. In this section, we will learn how to evaluate limits at infinity algebraically using the "Highest Power Rule", with tricks like using conjugates, common denominators, and factoring.

The line $y = L$ is a horizontal asymptote of the curve $y = f(x)$ in any of the following two cases:

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