Definition:Asymptotic Equality

Sequences
Let $\left \langle {a_n} \right \rangle$ and $\left \langle {b_n} \right \rangle$ be sequences in $\R$.

Then $\left \langle {a_n} \right \rangle$ is asymptotically equal to $\left \langle {b_n} \right \rangle$ iff $\dfrac {a_n} {b_n} \to 1$ as $n \to \infty$.

Functions
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 $g$.

Notation
The notation $a_n \sim b_n$ and $f \sim g$ is frequently seen to denote asymptotic equality.