Definition:Order of Infinitesimals

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Definition

Let $x$ and $y$ be real variables which tend to the limit $0$.

Then $x$ and $y$ are infinitesimals of the same order if and only if:

$\dfrac x y$ is finite and non-zero.

If $\dfrac x y \to 0$, then $x$ is an infinitesimal of higher order than $y$

If $\dfrac x y \to \infty$ or $\dfrac x y \to -\infty$, then $x$ is an infinitesimal of lower order than $y$.

Let $\dfrac x {y^n} \to 0$ is finite and non-zero, and $y$ is taken to be the first order.

Then $x$ is of $n$th order.

Also see

• Results about order of infinitesimals can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): order: 5. (of infinitesimals)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): order: 5. (of infinitesimals)