Limits at infinity - horizontal asymptotes - Limits

Limits at infinity - horizontal asymptotes

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.

Lessons

Notes:

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

i) $lim_{x \to\infty } f\left( x \right) = L$ ii) $lim_{x \to,-\infty } f\left( x \right) = L$

or

or

i) $lim_{x \to\infty } f\left( x \right) = L$	ii) $lim_{x \to,-\infty } f\left( x \right) = L$

or	or

Intro Lesson

Introduction to Horizontal Asymptotes
3.
Use "Highest Power Rule" to Evaluate Limits at Infinity of Rational Functions in 3 Types
Find:
5.
Multiply Conjugates First, then Evaluate Limits
Find:

Limits at infinity - horizontal asymptotes

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Lessons

Notes:

The line y=Ly = Ly=L is a horizontal asymptote of the curve y=f(x)y = f(x)y=f(x) in any of the following two cases: i)limx→∞f(x)=Llim_{x \to\infty } f\left( x \right) = Llimx→∞​f(x)=L ii) limx→,−∞f(x)=Llim_{x \to,-\infty } f\left( x \right) = Llimx→,−∞​f(x)=L or or

Intro Lesson

Introduction to Horizontal Asymptotes

a)

opposite relationship between "vertical asymptote" and "horizontal asymptote"

b)

how horizontal asymptotes are defined on each end of a function

c)

evaluate limits at infinity algebraically –"Highest Power Rule"!

d)

lesson overview

1.

Relate Asymptotes to Limits Express all asymptotes in limit notations for the function f ff whose graph is shown below.

2.

Discuss the Foundation of Limits at Infinity Find: i) limx→∞ 1xlim_{x \to \infty } \;\frac{1}{x}limx→∞​x1​ ii) limx→−∞ 1xlim_{x \to - \infty } \;\frac{1}{x}limx→−∞​x1​

3.

Use "Highest Power Rule" to Evaluate Limits at Infinity of Rational Functions in 3 Types Find:

a)

limx→∞ −5x2+13x+1002x2−8lim_{x \to \infty } \;\frac{{-5x^2+13x+100}}{{2x^2-8}}limx→∞​2x2−8−5x2+13x+100​ (Type 1: degree of numerator = degree of denominator)

b)

limx→−∞ 2x−9x3+7x2+10x+21lim_{x \to - \infty } \;\frac{{2x - 9}}{{{x^3} + 7{x^2} + 10x + 21}}limx→−∞​x3+7x2+10x+212x−9​ (Type 2: degree of numerator < degree of denominator)

c)

limx→∞ x2−3x+115−xlim_{x \to \infty } \;\frac{{{x^2} - 3x + 11}}{{5 - x}}limx→∞​5−xx2−3x+11​ (Type 3: degree of numerator > degree of denominator)

4.

5.

Multiply Conjugates First, then Evaluate Limits Find:

a)

limx→∞ (9x2+12x−3x)lim_{x \to \infty } \;\left( {\sqrt {9{x^2} + 12x} - 3x} \right)limx→∞​(9x2+12x​−3x)

b)

limx→−∞ (x+x2−5x)lim_{x \to - \infty } \;\left( {x + \sqrt {{x^2} - 5x} } \right)limx→−∞​(x+x2−5x​)

6.

Infinite Limits at Infinity Find: i) limx→∞ x3lim_{x \to \infty } \;{x^3}limx→∞​x3 ii) limx→−∞ x3lim_{x \to - \infty } \;{x^3}limx→−∞​x3

7.

Ambiguous Case: ∞−∞ \infty - \infty ∞−∞ Find limx→∞ x2−xlim_{x \to \infty } \;{x^2} - xlimx→∞​x2−x

8.

Limits at Infinity of Exponential Functions Find: i) limx→∞ exlim_{x \to \infty } \;{e^x}limx→∞​ex ii) limx→−∞ exlim_{x \to - \infty } \;{e^x}limx→−∞​ex

9.

Limits at Infinity of Trigonometric Functions Find limx→∞ sin⁡xlim_{x \to \infty } \;\sin xlimx→∞​sinx

Limits at infinity - horizontal asymptotes

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or

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

i) $lim_{x \to\infty } f\left( x \right) = L$ ii) $lim_{x \to,-\infty } f\left( x \right) = L$

or

or

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)

Multiply Conjugates First, then Evaluate Limits
Find:

$lim_{x \to \infty } \;\left( {\sqrt {9{x^2} + 12x} - 3x} \right)$

$lim_{x \to - \infty } \;\left( {x + \sqrt {{x^2} - 5x} } \right)$

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