Say I have a function $f$ for which $\lim_{x\rightarrow\infty}f(x)=\infty$ and which I'd like to approximate by a simpler function $g$. We say $g$ is an asymptotic for $f$ iff $ \lim_{x\rightarrow\infty} \frac{f(x)}{g(x)} = 1, $ i.e., iff the values $f(x)$ can be replaced with the values $g(x)$ with a percentage error that goes to $0$ as $x\rightarrow\infty$.
Sometimes the asymptotic $g(x)$ is even better, in the sense that the absolute error (not just the percentage error) goes to $0$ as $x\rightarrow\infty$, i.e., we have $ \lim_{x\rightarrow\infty}(f(x)-g(x)) = 0, $ or said another way, $ f(x) = g(x) + o(1). $
My question is this: Do such asymptotics have a special name? To give a silly example, consider $f(x)=x^2+1/x$. We have that $g(x)=x^2+x$ is an asymptotic for $f$, but it doesn't meet the second condition. A better asymptotic (one that does meet the second condition) would just be $g(x)=x^2$.