Can an asymptote be a curve? From what I have read, it suggest that the numerator must strictly be only one degree higher than than the denominator. However mathematically speaking, a equation like
$f(x) = \frac{x^3(x-1) + 1}{x-1}= x^3 + \frac{1}{x-1}$
would have an asymptote of $x^3$.
An asymptote can be any function at all, in fact! To say that the function $f(x)$ is asymptotic to the curve $g(x)$ just means that if $x$ is large then $f(x)$ and $g(x)$ are very close. Alternatively this means that $f(x) = g(x) + \varepsilon(x)$ where $\varepsilon$ is some function depending on $x$ that gets really small when $x$ is large. For instance, something like $\varepsilon(x) = x^{-1}$ or $\varepsilon(x) = e^{-x}$ would do the trick.
Here's a picture of a function with a curved asymptote. The green function is asymptotic to the red one.