Does $y= x^3$ have a horizontal asymptote at $x = 0$

Question

Does the function $f(x)=x^3$ have a horizontal asymptote at $x=0$?

I know that the derivative of $f(x)$ approaches $0$ as $x$ approaches $0$, so does that mean $y=0$ is a horizontal asymptote?

Answer 1

No. This is a matter of definition - a horizontal asymptote is not any point where the derivative is zero. Instead, we say a function has a horizontal asymptote of $y = c$ if $\lim_{x \to \infty}f(x) = c$ or $\lim_{x \to -\infty}f(x) = c$. Visually, this looks like $f(x)$ "flattening out" and approaching the line $y = c$ as you go off to the right or the left. The critical part of this is that it's a limit towards infinity, not towards a specific point.

A point where the derivative is zero is known as a critical point. On $f(x) = x^3$, the critical point at $x = 0$ is known as a saddle point, since it's neither a local minimum nor a local maximum.