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I am used to employing Maple in teaching Calculus. Today, I asked a student to think of finding the vertical asymptote(s) of a function for example, $y=\frac{x+1}{x^2-25}$ by Maple. I wanted her to work with the denominator in someway, but she solve the equation respect to $x$ and then apply the definition of horizontal asymptote for $y^{-1}$.

It looked interesting, but may I ask if this way is legal in calculus for finding vertical asymptote of a function? I want to be sure of this way. Of course, I think there are some certain conditions for a function in this question, like existing $y^{-1}$. Thanks for your any ides.

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    You have a very creative student! ;-)2013-04-20

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Since $ y=f(x) \Leftrightarrow x=f^{-1}(y), $ the graph of $f^{-1}$ is the graph of $f$ rotated so that the $x$-axis becomes the $y$-axis. Hence a horizontal asymptote for $f^{-1}$ becomes a vertical asymptote for $f$, and viceversa.

It is a nice approach, indeed. Of course it requires the invertibility of $f$, at least locally. But it is an interesting intuition for a student.

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    Thanks so much for making me sure. :)2012-11-02