Let be more precise: define an "strong elementary function" by only admitting rational for the "constant function" in the usual definition of "elementary" function (see for example: https://en.wikipedia.org/wiki/Elementary_function). Let $a$ be an "elementary real" if the constant function $f(x)=a$ is a strong elementary function. With this definition some non rational reals are elementary (for example $\pi = 4\cdot \arctan(1)$); but there are reals that are not elementary. Now let $f(x)$ be a strong elementary function defined in an open interval of an elementary real $a$ with the possible exception of $a$. Suppose that $$\lim_{x\rightarrow a} f(x)$$ exists. Is this limit necessarily an elementary real?
The idea behind this question is the following: when we learn to calculate limits in elementary calculus, it seems that there is always a method to do it. By calculating a limit we mean to define it by means of the rationals and elementary functions. But is there a general argument that prove that it is always possible?