Finding a Vertical AsymptoteDate: 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/ |
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