how to find a horizontal asymptote of a function?

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embizze | High School Teacher | (Level 1) Educator Emeritus

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A horizontal asymptote is a line y=k that the function approaches (gets arbitrarily close to) as x tends to positive or negative infinity.

(1) Rational functions: There are three types of rational functions; the degree of the numerator is less than, equal to, or greater than the degree of the denominator.

If the degree of the numerator is less than the degree of the denominator, then the function has the horizontal asymptote of y=0. For example if the function is `y=(x+3)/(x^2+3x+2)` then the horizontal asymptote is y=0.

If the degree of the numerator equals the degree of the denominator, then write both numerator and denominator in standard form. The horizontal asymptote is `y=a_m/b_m` where `a_m,b_m` are the leading coefficients of the numerator and denominator respectively. For example if `y=(2x^2+4x+3)/(3x^2-5x+2)` then the horizontal asymptote is `y=2/3` :

If the degree of the numerator is greater than the degree of the denominator then there is no horizontal asymptote. (Though there may be a slant asymptote.)

(2) Other functions have horizontal asymptotes. Logistics functions, certain trigonometric inverse functions, and compositions of some functions will have horizontal asymptotes. Typically we use limits to find the asymptotes for these cases, though there may be short-cut methods. (e.g. for logistics functions of the form `y=c/(1+ae^(-bx))` where `e` is the exponential (Euler's constant), then the asymptotes are y=0 and y=c.)

Note that a function can have no more than 2 horizontal asymptotes. Also a common misperception is that a function cannot cross its horizontal asymptote -- consider `y=2+1/xcos(x)` ; this function has a horizontal asymptote of y=2 and crosses this line an infinite number of times:

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alejandrogalarce | Student, Undergraduate | (Level 1) Valedictorian

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Whereas vertical asymptotes are sacred ground, horizontal asymptotes are just useful suggestions. Whereas you can never touch a vertical asymptote, you can (and often do) touch and even cross horizontal asymptotes. Whereas vertical asymptotes indicate very specific behavior (on the graph), usually close to the origin, horizontal asymptotes indicate general behavior far off to the sides of the graph. To get the idea of horizontal asymptotes, let's looks at some simple examples.

  • Find the horizontal asymptote of the following function:

The horizontal asymptote tells me, roughly, where the graph will go when x is really, really big. So I'll look at some very big values for x, some values of x very far from the origin:


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