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?
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 $y = \dfrac{x^3}{x^3}$.