Definition:Asymptotic Equality

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

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

Functions
Let $$f$$ and $$g$$ real functions defined on $$\mathbb{R}$$.

Then $$f$$ is asymptotically equal to $$g$$ iff $$\frac {f \left({x}\right)} {g \left({x}\right)} \to 1$$ 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.