# Definition:Asymptotic Equality/Sequences

## Definition

Let $\sequence {a_n}$ and $\sequence {b_n}$ be sequences in $\R$.

### Definition 1

Let $b_n \ne 0$ for all $n$.

$\sequence {a_n}$ is asymptotically equal to $\sequence {b_n}$ if and only if:

$\ds \lim_{n \mathop \to \infty} \dfrac {a_n} {b_n} = 1$

### Definition 2

$\sequence {a_n}$ is asymptotically equal to $\sequence {b_n}$ if and only if:

$a_n - b_n = \map o {b_n}$

where $o$ denotes little-$\oo$ notation.

### Definition 3

$\sequence {a_n}$ is asymptotically equal to $\sequence {b_n}$ if and only if:

$a_n - b_n = \map \oo {a_n}$

where $\oo$ denotes little-$\oo$ notation.

This is denoted: $a_n \sim b_n$.