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I am a graduate student studying math, and am actually teaching College Algebra right now. But every once in a while, I come upon something new in a subject that I have supposedly mastered.

Why does the graph of $$y=\frac{x^3}{x^3}$$ not have a horizontal asymptote at $y=1$? Or does it? What am I missing?

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    Possible typo? The function $y = y(x)$ is not defined for $x = 0$, and has value of $1$ everywhere else. – Ivo Terek Jul 23 '14 at 00:29
  • Apart from the removable singularity at $x=0$, $y=x^3/x^3$ is that asymptote... – Semiclassical Jul 23 '14 at 00:30
  • The graph is basically a horizontal asymptote at $y = 1$, apart from when $x = 0$ – Vishwa Iyer Jul 23 '14 at 00:31
  • I feel as though this is quite a confusing question but I'd agree with the other replies. I'd also say that the graph doesn't approach $y=1$ asymptotically (but is already touching it) because $y$ identically $1$ for all nonzero $x$. – Jam Jul 23 '14 at 00:34
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    Ok. I guess I was missing the knowledge that a rational function can intersect its horizontal asymptote. –  Jul 23 '14 at 00:35

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It depends on what your definition of a horizontal asymptote is.

The calculus book I've used says that the graph of $y = f(x)$ has a horizontal asymptote at $y = b$ iff $\displaystyle\lim_{x \to \infty}f(x) = b$ or $\displaystyle\lim_{x \to -\infty}f(x) = b$.

In this case, $\displaystyle\lim_{x \to \infty}\dfrac{x^3}{x^3} = \lim_{x \to \infty}1 = 1$, so $y = \dfrac{x^3}{x^3}$ has a horizontal asymptote at $y = 1$.

Wikipedia's article on Asymptote says that "an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity. Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors." If you don't include that requirement, then $y = 1$ is a horizontal asymptote of $y = \dfrac{x^3}{x^3}$.

JimmyK4542
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  • While I was intrigued to discover that old and "unusual" requirement, and even more intrigued that it is rooted in the original meaning of the word asymptote, nonetheless I think the Wikipedia page has erred in giving that old meaning such prominence. The word is now used just as you have written; that old prohibition is completely gone in modern usage. – Lee Mosher Jul 23 '14 at 05:00