Is there a term that means f(x) approaches an asymptote while remaining on the same side? Limit of $\frac{1}{x}$ as x approaches infinity is zero, and so is the limit of $\frac{\sin(x)}{x}$ as $x$ approaches infinity. However, as far as I know we don't have a concise way to say $\frac{1}{x}$ approaches the real axis as $x$ approaches infinity and as $x$ is a finite number approaching infinity $\frac{1}{x}$ is always positive. On the other hand, $\frac{\sin(x)}{x}$ does not poses this property. Wouldn't it be useful to define a word that means $f(x)$ approaches a certain asymptote as $x$ approaches infinity, and is always on the same side of the asymptote? I think we should also have a convenient way to say from which side it approaches the asymptote. Do we already have a way to do that?
 A: It might not be the specific terminology you're looking for in this particular context, but:


*

*for the $\tfrac{\sin x}{x}$-behaviour, you could say the function tends to its aymptote oscillatorily;

*for the $\tfrac{1}{x}$-behaviour, you could say the function tends tends to its aymptote monotonically (decreasing, in this case).

A: You sometimes see $\searrow$ or $\downarrow$ used instead of $\rightarrow$ to indicate that a function or sequence decreases to its limit rather than merely approaches it.  But usually words are coined as needed, not because they'd be nice to have.  It could be there aren't enough theorems about the functions you describe that a whole new word is needed to describe them.  Or, perhaps, the combination of limit and monotonicity is enough.
A: I often used, in my classes, the term $0^+$ to indicate that $f$ was approaching $0$ by positive values.
This would be common in my classes: $\displaystyle\lim_{x\to 0^+} \frac{1}{1 - 1/x} = 0^-$ or any variation of it.
However, this was quite informal and even though my teachers would write it on the board, I don't know if it's common or widely accepted
A: If the function $f(x)$ is greater (less than) than its asymptote $L$, you say $f(x)$ approaches $L$ from above (below) as $x \to$ whatever. 
