When $y=f(x)+ax+b$ for some trig function $f$, is there a name (like "oblique asymptote") for the line $y=ax+b$? 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 $f(x):=\sin(x)+x$ :

 A: 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).
A: The definition of an asymptote is a function that your function approaches as the limit goes to infinity in one direction or the other in other words $f(x)$ is asymptotic to $g(x)$ as $\lim_{x\to \infty}(f(x)-g(x))=0$.   In this case,  the limit fails to converge as the limit is just your trig function, so asymptote is very much not the right language.
I'm not sure if there does exist a term for what you want.
