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### Finding a Vertical Asymptote

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Date: 05/11/2000 at 13:32:54
From: Audrey Connor
Subject: Calculus - finding vertical asymptote

In a calculus problem, one task was to find the vertical asymptote of
the equation:

xy^2 - x^3y = 6

I tried, and thought maybe it was x = sqrt(2y), but that doesn't seem
to make sense, and I can't seem to get a sense of what the graph looks
like.
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Date: 05/11/2000 at 16:12:46
From: Doctor Rob
Subject: Re: Calculus - finding vertical asymptote

Thanks for writing to Ask Dr. Math, Audrey.

Write the equation in the form

y*(y-x^2) = 6/x.

Near a vertical asymptote x is bounded and y grows without bound, so
the left side grows without bound. That implies that 6/x grows without
bound. That happens if and only if x approaches zero. That means that
there is a vertical asymptote at x = 0, and it is the only vertical
asymptote.

A general way to find asymptotes is as follows. Start with the
equation and discard all terms whose degree is not the same as the
highest degree term, bring all terms to one side, and factor what you
have. Set any first-degree factors equal to zero, and you'll have
equations that give you the slopes of the asymptotes. If one of these
looks like:

A*x + B*y = 0

then there is an asymptote with slope m = -A/B. The equation of the
asymptote will have the form:

A*x + B*y = C

Use this to eliminate one of the variables from the original equation
by substitution. Take the term with the highest power of the remaining
variable, and set its coefficient equal to zero. Solve the resulting
equation for C, and you have the equation of the asymptote.

In your case, you started with:

x*y^2 - x^3*y - 6 = 0

The highest degree term is -x^3*y, which has linear factors x and y.
Then the equations of the asymptotes will have the form:

0*x + 1*y = C
or
1*x + 0*y = C

that is, y = C or x = C. Thus in this case, the asymptotes are either
horizontal or vertical. For y = C, substituting, you get:

-C*x^3 + C^2*x - 6 = 0

The highest power of x is x^3, so we set its coefficient -C = 0, and
solve, getting C = 0. Thus the horizontal asymptote is y = 0.

For x = C, substituting, you get:

C*y^2 - C^3*y - 6 = 0

The highest power of y is y^2, so we set its coefficient C = 0, and
solve, getting C = 0. Thus the vertical asymptote is x = 0.

- Doctor Rob, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
College Calculus
High School Calculus

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