Can asymptotes be curved?

When I was first introduced to the idea of an asymptote, I was taught about horizontal asymptotes (of form y=a) and vertical ones ( of form x=b).

I was then shown oblique asymptotes-- slanted asymptotes which are not constant (of the form y=ax+b).

What happens, though, if we've got a function such as

$f(x)={e}^{x}+\frac{1}{x}?$

Is $y={e}^{x}$ considered an asymptote in this example?

Another example, just to show you where I'm coming from, is

$g(x)={x}^{2}+\mathrm{sin}(x)$

-- is $y={x}^{2}$ an asymptote in this case?

The reason that I ask is that I don't really see the point in defining oblique asymptotes and not curved ones; surely, if we want to know the behaviour of y as $x\to \mathrm{\infty}$, we should include all types of functions as asymptotes.

If asymptotes cannot be curves, then why arbitrarily restrict asymptotes to lines?

When I was first introduced to the idea of an asymptote, I was taught about horizontal asymptotes (of form y=a) and vertical ones ( of form x=b).

I was then shown oblique asymptotes-- slanted asymptotes which are not constant (of the form y=ax+b).

What happens, though, if we've got a function such as

$f(x)={e}^{x}+\frac{1}{x}?$

Is $y={e}^{x}$ considered an asymptote in this example?

Another example, just to show you where I'm coming from, is

$g(x)={x}^{2}+\mathrm{sin}(x)$

-- is $y={x}^{2}$ an asymptote in this case?

The reason that I ask is that I don't really see the point in defining oblique asymptotes and not curved ones; surely, if we want to know the behaviour of y as $x\to \mathrm{\infty}$, we should include all types of functions as asymptotes.

If asymptotes cannot be curves, then why arbitrarily restrict asymptotes to lines?