The answer is probably different, depending on what you mean by the limit.
(Existing of a limit as $x\to a$ when $x$ is complex is a much stronger condition then existence of a limit when $x$ is real). 

1. Complex limits. The answer seems to be yes. What follows is a heuristic argument. Elementary functions are analytic, with at most countably many singularities.
So when the finite limit exists, the singularity is removable or a ramification point. Expanding everything in power series (perhaps with fractional powers), we can compute the limit. The coefficients of these power series are elementary constants, because they are expressed in terms of derivatives. So the answer is probably yes, if we mean complex limits. (This is heuristic because singularities can accumulate so one needs more careful argument). 

2. Real limits. The answer seems to be no. Consider a trigonometric sum 
$$f(t)=\sum_{j=1}^n a_j\exp(\lambda_jit),\quad i=\sqrt{-1},$$
and assume that it has no real zeros. Suppose that the $\lambda_j$ are real "elementary constants" but incommensurable. Then $f(t)=r(t)\exp(i\phi(t))$, where $r(t)>0$ and $\phi$ is a well-defined elementary real function. The limit
$$m:=\lim_{t\to+\infty}\phi(t)/t$$ always exists: this is the celebrated Mean Motion Theorem. There is a formula for this limit due to A. Wintner:
$$m=\sum_{j=1}^n\lambda_jW_j,\quad\mbox{where}\quad W_j=\int_{T^n}\Re\frac{a_j\exp(i\theta_j)}{\sum_ka_k\exp(i\theta_k)}d\theta_1\ldots d\theta_n,$$
where $T^n=[0,2\pi]^n$.
These integrals are probably not "elementary constants": they can be expressed 
in Bessel functions. Again, this argument is incomplete, because one has to
prove that the $W_j$ can indeed be "non-elementary" constants. (Of course you can replace $t\to\infty$ by $t\to 0$ by changing the variable to $1/t$. The difference between case 1 and case 2 is that a function can have an essential singularity and still the real limit exists).

To complete this answer one has to find specific $a_j$, evaluate the integral in terms of Bessel functions, and show that the answer is indeed not an elementary constant. This can be difficult; it is not clear what the set of elementary constants is, and how to prove that some number is not an "elementary constant".

References on the mean motion theorem: S. Sternberg, Celestial Mechanics, Part I, Benjamin, 1969, and on the expression of $W_j$ in Bessel functions: G. Watson, Treatease on Bessel functions, CUP, 1958.