MATH 12002 - CALCULUS I §1.6: Horizontal Asymptotes

Professor Donald L. White

Department of Mathematical Sciences Kent State University

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Horizontal asymptotes are very closely related to limits at infinity. Definition Let y = f (x) be a function and let L be a number. The line y = L is a horizontal asymptote of f if

lim f (x) = L or lim f (x) = L. x→+∞ x→−∞

Notes: The definition means that the graph of f is very close to the horizontal line y = L for large (positive or negative) values of x. A function can have at most two different horizontal asymptotes. A graph can approach a horizontal asymptote in many different ways; see Figure 8 in §1.6 of the text for graphical illustrations. In particular, a graph can, and often does, cross a horizontal asymptote. Let’s emphasize this point to avoid problems later:

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THERE’S NO REASON — HERE’S MY POINT, DUDE — THERE’S NO #∧♣%!*& REASON WHY THE GRAPH OF A FUNCTION CAN’T CROSS A HORIZONTAL ASYMPTOTE!!

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Our general result on limits at infinity of rational functions implies the following. Theorem P(x) Let f (x) = Q(x) be a rational function (P(x),Q(x) are polynomials). If deg P(x) = deg Q(x), then the horizontal asymptote of f (x) is p y = , q

where p, q are the leading coefficients of P(x),Q(x), respectively. If deg P(x) < deg Q(x), then the horizontal asymptote of f (x) is y = 0 (the x-axis). If deg P(x) > deg Q(x), then f (x) has no horizontal asymptote.

Note: In general, a function need not have the same horizontal asymptote at both −∞ and +∞. (See Exercise 1.6.34 in the text, for example.) D.L. White (Kent State University) 4 / 5 Examples

1 The rational function 4x3 − 7x2 + 2 f (x) = 5 − 9x3 has both numerator and denominator of degree 3. The numerator has leading coefficient 4, and the denominator has leading coefficient −9. Therefore, the horizontal asymptote (at both +∞ and −∞) is 4 4 y = = − . −9 9 2 The rational function 7x4 + 3x2 + 5 f (x) = 6x5 − x has numerator of degree 4 and denominator of degree 5. Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0.

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