TITLE: When $y=f(x)+ax+b$ for some trig function $f$, is there a name (like "oblique asymptote") for the line $y=ax+b$?
QUESTION [5 upvotes]: Let $f$ be a trig function, and let $y:=ax+b$ be a linear function. Every time you have $f(x)+y$ as one function, you get a graph where $(C_f)$ follows the line $y=ax+b$, but we don't have the known limit for an oblique asymptote ($\lim_{x \rightarrow \infty} f(x)-y \neq 0 $).
Can we still call the line $y:=ax+b$ an oblique asymptote, or does this line have another name?
An example with the function $f(x):=\sin(x)+x$ :

REPLY [2 votes]: The function $x \mapsto \sin(x) - x$ does not have an oblique asymptote...
The answer to your question is that it all comes down to definitions:  what, precisely, is the definition of the term "asymptote"?  Wikipedia gives a definition which seems fairly workable:

Definition:  In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the $x$ or $y$ coordinates tends to infinity.

MathWorld gives a similar definition:

Definition: An asymptote is a line or curve that approaches a given curve arbitrarily closely.

There is also a definition in Thomas' Calculus (13th ed, p. 91), though this definition is specifically related to the oblique asymptotes of a rational function:

Definition:  If the degree of the numerator of a rational function $f$ is $1$ greater than the degree of the denominator, the graph has an oblique or slant line asymptote.  We find an equation for the asymptote by dividing numerator by denominator to express $f$ as a linear function plus a remainder term. that goes to zero as $x \to \pm \infty$.

While these are little informal (or, perhaps, either too general or specific for the example given in this question), the essential idea is that a line is an asymptote of a curve if the curve can be made arbitrarily close to the line in "the limit".  Perhaps more precisely, if $f$ and $\ell$ functions such that
$$ \ell(x) = ax + b, $$
then the line $\ell$ is an asymptote of $f$ if either
$$ \lim_{x\to \infty} f(x) - \ell(x) = 0
\qquad\text{or}\qquad
\lim_{x\to-\infty} f(x) - \ell(x) = 0. $$
The crux of the definitions is that the "gap" between the two curves can be made arbitrarily small by choosing $x$ to be "large enough".
In the example given,
$$ f(x) = \sin(x) + x$$
and the potential asymptote is
$$ \ell(x) = x. $$
Observe that
$$\lim_{x\to \pm\infty} f(x) - \ell(x)
= \lim_{x\to\pm \infty} (\sin(x)+x) - x
= \lim_{x\to\pm\infty} \sin(x), $$
which does not exist (the limit does not converge; the function oscillates).  Therefore $\ell$ fails to be an asymptote of $f$.
...but it is "on the order of" $x$.
On the other hand, if one "zooms out" far enough, $f$ and $\ell$ seem to exhibit similar behaviour—the two functions are "asymptotically similar" in some sense.  The usual way to make sense of this intuitive idea is through Landau notation.  The idea here, I think, is that of asymptotic similarity:

Definition: A function $f$ is asymptotically similar to (or on the order of) $g$ (at infinity), denoted $f \sim g$, if
$$ \lim_{x\to \infty} \left| \frac{f(x)}{g(x)} \right| = 1.$$

In the case of the example given,
\begin{align}
\lim_{x\to \infty} \left| \frac{\sin(x) + x}{x} \right|
&\le \lim_{x\to \infty} \frac{|\sin(x)|+|x|}{|x|} && \text{(triangle inequality)} \\
&= \lim_{x\to\infty} \frac{|\sin(x)|}{x} + \lim_{x\to\infty} \frac{x}{x} \\
&= 1,
\end{align}
therefore $\sin(x) + x \sim x$; that is, $\sin(x) + x$ is on the order of $x$ when $x$ is large.
Note that it is also possible to talk about a function $f$ being on the order of $g$ "at a point"—consider the limit as $x\to a$ rather than $x\to \infty$.  Also note that other notions of asymptotic behaviour are characterized by in other ways with other symbols (e.g. big-Oh notation, little-oh notation, and so on).  These are nicely summarized by Wikipedia (this is the same link as above).