Finding a Vertical Asymptote

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.

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/