Let's call $C$ the set of computable real numbers and $L$ the set of all the (existing) limits of finite length algebraic expressions.
By $L$ I mean the set of all converging limits $\lim_{x_1 \to \infty} \lim_{x_2 \to \infty} \cdots \lim_{x_n \to \infty} f(x_1,x_2,\dots,x_n)$, where $n \in \mathbb{Z}^{+}$, and $f$ is a function that may be written as a finite length expression that includes only algebraic numbers, the variables $x_1,x_2,...,x_n$ and the folowing operators: $+,-,\cdot,\div,\sum$ and $\prod$.
It's obvious that $L \subseteq C$ because if a real number can be writen as a limit, it may be approximatted as much as desired.
So the question is: $C \subseteq L$ holds?, which may be rewritten as: is there a computable number which is not the limit of any finite length algebraic expression?
If it does exist, can anybody show me such $c\in C$ with $c\notin L$?