- 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.

- 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.

8.5.2 Horizontal Asymptotes and Infinite Limits Flashcards by Irina Soloshenko

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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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.

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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.

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

y = 0

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

No horizontal asymptote exists.

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

y = 0

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

No horizontal asymptote exists

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Find the horizontal asymptote(s) given

f(x) = 3x^2+4x / 2x^2−1

y = 3/2

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

y = −1/3, y = 1/3

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8.5.2 Horizontal Asymptotes and Infinite Limits Flashcards

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