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### Dominant Terms

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Date: 03/25/98 at 14:31:44
Subject: Dominant terms

Dear Dr. Math,

I am a senior in Calculus and we are working on the first and second
derivative tests for increasing/decreasing values and concavity. We
are now confronted with using these tests along with asymptotes and
dominant terms to graph functions. I am confused on what dominant
terms are and how to obtain their values, and once the values are

An example problem:

f(x) = (x^4+1)\(x^2)

For the function, what are the dominant terms and the asymptotes?

Thank you for your time and effort. And thanks to the Math Forum for
setting up this valuable tool.

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Date: 03/25/98 at 15:39:41
From: Doctor Rob
Subject: Re: Dominant terms

A dominant term such as x -> a is a term which grows more rapidly than
any other in the expression. In your example there are two terms:

f(x) = x^2 + 1/x^2.

As x -> infinity, the dominant term is x^2, because the ratio of x^2
to 1/x^2 (which equals x^4) goes to infinity. The function behaves
like x^2 for large values of x, because the other terms are of
negligable size compared to the dominant one.

As x -> 0, the dominant term is 1/x^2, because the ratio of 1/x^2 to
x^2 (which equals 1/x^4) goes to infinity. The function behaves like
1/x^2 for small values of x, because the other terms are of negligible
size compared to the dominant one.

There is a vertical asymptote at x = 0, as you might guess, because
x^2 appears in the denominator. Vertical asymptotes occur where the
numerator approaches infinity or the denominator approaches zero as
x -> a, a finite value. If this condition holds, the equation of the
asymptote is x = a.

There are no other asymptotes. If there were a nonvertical asymptote,
then f(x)/x would approach its slope, but f(x)/x approaches infinity
as x -> infinity. If lim f(x)/x = m exists, and if b = lim [f(x)-m*x],
then the equation of the nonvertical asymptote is y = m*x + b.

-Doctor Rob,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/
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Associated Topics:
High School Calculus

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