Is there standard terminology to describe the not-quite-a-limit behavior of ${\tan( \log x) \over x}$ as $x$ approaches infinity? Suppose I want to describe the long term behavior of ${\tan(\log x) \over x}$ as x increases towards positive real infinity.
Now, 
$$\lim_{x \rightarrow \infty}{\tan(\log x) \over x}$$
obviously doesn't exist.  So it would be wrong to say its limit is 0.
But in some very slightly looser sense, the term obviously approaches 0 aside from the very occasional vertical asymptote.  If you were to pick a point at "random" far, far down the number line (I'm being very imprecise here, I know), it would be an $\epsilon$ from 0 with a probability approaching 1 as the random range you were pulling from got larger.  Various summability methods would also make this fact clear.
Is there either standard terminology for getting this idea across, or standard notation for expressing it?

It's not clear to me how wide the "bad" intervals are.  If they are sufficiently narrow, then we could say the functions $$f_n(x)=\chi_{[n,\infty)}\frac{\tan(\log x)}{x}$$
converge to the constant zero function $f(x)\equiv 0$ globally in measure.  That would mean that the total measure of the "bad" intervals is bounded, and goes to zero as $n\to \infty$ (that is, as we ignore an increasingly larger initial segment of the $\frac{\tan(\log x)}{x}$ bit).
