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AP Calculus AB: 8.5.2 Horizontal Asymptotes and Infinite Limits

Mathematics9 CardsCreated 7 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.

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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.

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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

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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