CramX
RegisterLog in
Back to AI Flashcard MakerMathematics /AP Calculus AB: 8.5.2 Horizontal Asymptotes and Infinite Limits

AP Calculus AB: 8.5.2 Horizontal Asymptotes and Infinite Limits

Mathematics9 CardsCreated 3 months ago

This content explains how to identify horizontal asymptotes by evaluating the limits of functions as x approaches positive or negative infinity. It focuses on comparing the highest powers in the numerator and denominator of rational functions to determine the behavior and horizontal leveling of the graph.

PrintEmbedImportReport

Horizontal Asymptotes and Infinite Limits

  • Asymptotes are lines that the graph of a function approaches. A horizontal asymptote to the graph of a function f is a line whose equation is y = a

  • Identify horizontal asymptotes by taking the limit of the function as x approaches positive or negative infinity.

  • Examine the highest-powered term in the numerator and the highest-powered term in the denominator when determining the limit of a rational function. The expression is an indeterminate form.

Tap or swipe ↕ to flip
Swipe ←→Navigate
1/9

Key Terms

Term
Definition

Horizontal Asymptotes and Infinite Limits

  • Asymptotes are lines that the graph of a function approaches. A horizontal asymptote to the graph of a function f is a line whose equation ...

note 1

  • A horizontal asymptote is present when the graph of a function levels off at positive infinity or negative infinity. Because it is a horizo...

note 2

  • When evaluating the limit of a rational function at infinity, it is useful to ask the question “Which part is approaching infinity faster?”...

Find the horizontal asymptote(s) of f(x). f(x)= |x| /2x^2+2

y = 0

Find the horizontal asymptote(s) given f(x) = 3x+2.

No horizontal asymptote exists.

Find the horizontal asymptote(s) given f(x) = 1/3x.

y = 0

Log in to view all terms

Related Flashcard Decks

Mathematics

Essential SAT Math Formulas

This deck covers key formulas and concepts essential for the SAT Math section, including slope, distance, and vertex forms.

10 cards
View Deck
Mathematics

Algebra 1 Unit 1 Mastery Test

These properties of equality help maintain balance in an equation when performing the same operation on both sides. Inverse operations, like using addition to undo subtraction, are key tools in solving equations.

29 cards
View Deck
Mathematics

ATI TEAS Test Practice Mathematics Part 2

This flashcard set includes a variety of practical math questions covering percentages, conversions, averages, scale drawings, and basic algebra. It’s designed to strengthen problem-solving skills for real-life and standardized test situations.

53 cards
View Deck
Mathematics

ATI TEAS Test Practice Mathematics Part 1

This flashcard set includes a variety of practical math questions covering percentages, conversions, averages, scale drawings, and basic algebra. It’s designed to strengthen problem-solving skills for real-life and standardized test situations.

55 cards
View Deck
Mathematics

Unit 2 Progress Check: MCQ Part A

To determine the point where the tangent line to the graph of the function 𝑓f has a slope of 2 for 𝑥 >0 x>0, we use the fact that the slope of the tangent line at any point is given by the derivative 𝑓′(𝑥)f′(x).

15 cards
View Deck
Mathematics

Customary Liquid Measurement Units

This set of flashcards outlines key relationships between customary units of liquid measurement, including pints, cups, quarts, and gallons. It provides conversion facts to help understand how these units relate to one another in the U.S. measurement system.

3 cards
View Deck

Study Tips

  • Press F to enter focus mode for distraction-free studying
  • Review cards regularly to improve retention
  • Try to recall the answer before flipping the card
  • Share this deck with friends to study together
AP Calculus AB: 8.5.2 Horizontal Asymptotes and Infinite Limits
TermDefinition

Horizontal Asymptotes and Infinite Limits

  • Asymptotes are lines that the graph of a function approaches. A horizontal asymptote to the graph of a function f is a line whose equation is y = a

  • Identify horizontal asymptotes by taking the limit of the function as x approaches positive or negative infinity.

  • Examine the highest-powered term in the numerator and the highest-powered term in the denominator when determining the limit of a rational function. The expression is an indeterminate form.

note 1

  • A horizontal asymptote is present when the graph of a function levels off at positive infinity or negative infinity. Because it is a horizontal line, the equation will be of the form y = a.

  • To understand how the function is behaving at infinity you need to take its limit at infinity. This will empower you to identify horizontal asymptotes.

  • To take a limit at infinity ( ), you need to recall that infinity represents a progression of increasing values. In this case, as x goes to (or gets larger and larger), 1/x goes to 0. Therefore, the limit of the function at is 0, and there is a horizontal asymptote at y = 0.

  • This chart shows that as x gets increasingly large, x 3 gets increasingly large, too. Therefore, the limit of the function at is . There is no horizontal asymptote. The function never levels off away from the origin.

note 2

  • When evaluating the limit of a rational function at infinity, it is useful to ask the question “Which part is approaching infinity faster?”

  • Higher powers will approach infinity faster than lower
    powers. Identify the highest-powered term in the numerator and compare it to the highest-powered term in the denominator. In this case, the denominator approaches infinity faster, so the limit is 0. Therefore, there is a horizontal asymptote at y = 0.

  • Here, the highest-powered term resides in the numerator. In the denominator, the x-term and the constant do not contribute much to the behavior of the function at infinity. You can actually ignore them and concentrate on the 5x 3 and –2x 2 terms to find the limit. After simplifying the new fraction, you can see that the numerator is going to infinity, but the denominator is negative. Therefore, the limit equals – . There is no horizontal asymptote.

  • To evaluate this limit you must consider the highest-powered term in the numerator and denominator. For the sake of the limit, you can ignore the other terms. Notice that when you cancel the common factors of x 2 , the result is the limit of 1/3 as x approaches . The limit of 1/3 is 1/3.

  • You can conclude the existence of a horizontal asymptote at y = 1/3.

  • Here is a generalization of the results above. You do not need to memorize them in order to evaluate limits at infinity.

Find the horizontal asymptote(s) of f(x). f(x)= |x| /2x^2+2

y = 0

Find the horizontal asymptote(s) given f(x) = 3x+2.

No horizontal asymptote exists.

Find the horizontal asymptote(s) given f(x) = 1/3x.

y = 0

Find the horizontal asymptote(s) given f(x) = x^2/9x+1

No horizontal asymptote exists

Find the horizontal asymptote(s) given

f(x) = 3x^2+4x / 2x^2−1

y = 3/2

Find the horizontal asymptote(s) given f(x) = |x| / 3x+1.

y = −1/3, y = 1/3