Are $x^2$ and $x^2 + 1$ asymptotic?

Question

You can see that as the curves go towards infinity, they come pretty close to each other, with the distance between them decreasing continuously. But I haven't seen any source saying they're asymptotic. Even my math teacher says it's an exception. So are they actually asymptotic or they're a mathematical exception?

Answer 1

They are asymptotically equivalent.

$$\lim_{x \to \infty} \frac{x^2+1}{x^2} = \lim_{x \to \infty}1+\frac{1}{x^2} = 1$$

In general, two nonzero polynomials are asymptotically equivalent if and only if they have the same leading coefficient and the same degree.