= Slowly varying function
{title2=$\ell(ut)/\ell(t)\to1$}
{wiki}
= Slowly varying
{synonym}
A positive <measurable function> $\ell$ is slowly varying at infinity if
$$
\frac{\ell(ut)}{\ell(t)}\longrightarrow1
\qquad(t\to\infty)
$$
for every fixed $u>0$. It has <regular variation> of index zero. “Slowly” concerns relative change under multiplication of the argument; it does not mean that the <function> must be bounded, have a small <derivative>, or converge to a finite value.
Every positive constant is a <slowly varying function>. So are $\log t$, $1/\log t$ and $(\log t)^a$ for any fixed <real number> $a$, on $t>1$. Indeed,
$$
\frac{(\log(ut))^a}{(\log t)^a}
=\left(1+\frac{\log u}{\log t}\right)^a\longrightarrow1.
$$
The first grows without bound, the second tends to zero, and the constant does neither. Another unbounded example is $\ell(t)=e^{\sqrt{\log t}}$:
$$
\log\frac{\ell(ut)}{\ell(t)}
=\frac{\log u}{\sqrt{\log t+\log u}+\sqrt{\log t}}
\longrightarrow0.
$$
This <function> can grow much faster than $\log t$ while retaining <regular variation> of index zero.
Even convergence is unnecessary. The bounded <function>
$$
\ell(t)=\exp\!\bigl(\sin(\log\log t)\bigr),\qquad t>1,
$$
oscillates between $e^{-1}$ and $e$. It is nevertheless a <slowly varying function>: the change in the argument of the <sine> under $t\mapsto ut$ is
$$
\log\log(ut)-\log\log t
=\log\left(1+\frac{\log u}{\log t}\right)\longrightarrow0,
$$
and the <Lipschitz continuity> of the <sine> makes the logarithm of the ratio tend to zero. The <sequences> $t_k=\exp(\exp(\pi/2+2\pi k))$ and $s_k=\exp(\exp(3\pi/2+2\pi k))$ give the two values $e$ and $e^{-1}$ for every nonnegative <integer> $k$, so there is no <limit> at infinity.
Contrast this with $h(t)=2+\sin(\log t)$, which is positive and bounded but is not a <slowly varying function>. Fix $u=e^{\pi/2}$. Along $t_k=e^{2\pi k}$ its rescaling ratio is $h(ut_k)/h(t_k)=3/2$; along $s_k=e^{\pi+2\pi k}$ the ratio is $1/2$. The persistent oscillations prevent even a <limit> for this one fixed rescaling. It is not <regularly varying> with any finite index either.
\Image[/slow-variation-oscillation-comparison.png]
{title=Two bounded oscillating <functions>: logarithmic rescaling makes one ratio approach one, while the other keeps oscillating}
{description=The horizontal coordinate is $s=\log t$, and the fixed rescaling is $u=e^{\pi/2}$. The displayed range illustrates the formulas; the limiting claims follow from the arguments in the text.}
The useful distinction is whether a fixed proportional change of input eventually produces a negligible proportional change of output. A <slowly varying function> can therefore preserve the scaling exponent of a <power law> while changing its magnitude substantially. The decomposition in <regular variation> explains why a logarithmic factor can matter in numerical estimates without changing the limiting exponent.
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