Can a curve be an asymptote?

Question

$f(x)=x^3+\frac{3}{x-1}$

This was the question given to me. I replied that $f(x)$ will have only a single vertical asymptote of $x=1$.

My teacher told that there'll be be two asymptotes. One is the vertical one($x=1$) and another is the curve $y=x^3$.

I checked on the Internet and except for Mathworld, which includes curve in its definition of asymptotes, every other site defines asymptote as a line.

Can a curve be an asymptote?

Answer 1

Our definition of asymptote:

Let $f$ be a real function defined on some neighborhood of $\infty$ and $a, b \in \mathbb{R}$. We say the function $ax + b$ is the asymptote of that function if $$\lim_{x \to\infty} (f(x) - ax - b) = 0.$$

(The definition of asymptote in $-\infty$ is analogous.)

In our definition, only the affine function $ax + b$ can be asymptote. What you call an asymptote (vertical line $x = 1$) we do not call asymptote at all.

That being said, it is a matter of definition and what you need this for. At high school during precalculus lessons, we used to call an asymptote everything "the function value tends to at the function plot".