Slowly varying function

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In real analysis, a branch of mathematics, a slowly varying function is a function resembling a function converging at infinity.

[edit] Definition

The function $\begin{align}&\scriptstyle L \colon (0,\infty) \to (0,\infty) \end{align}$ is slowly varying (at infinity) iff

$(\forall a>0): \lim_{x \to \infty} \frac{L(ax)}{L(x)}=1.$

[edit] Examples

If $\lim_{x \to \infty} L(x) = b \in (0,\infty)$ then $L$ is a slowly varying function.
For any $\beta \in \Bbb{R}$ , $L (x): = (log x) β$ is slowly varying.

[edit] References

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