Definition:Asymptotic Equality/Real Functions/Infinity

Definition
Let $f$ and $g$ real functions defined on $\R$.

Then $f$ is asymptotically equal to $g$ iff $\dfrac {f \left({x}\right)} {g \left({x}\right)} \to 1$ as $x \to +\infty$.

That is, the larger the $x$, the closer $f$ gets (relatively) to $g$.