Stone-Čech compactifications and limits of sequences
EDIT: As pointed by Stefan H. in his comment, the solution I have suggested only works if $X$ is normal, since I am using Tietze extension theorem.
Perhaps I have overlooked something and I will be blushing, but I will give it a try. (This is my solution, I did not check the books I mentioned in the above comment. Perhaps the proofs from those books can give you a hint for a different proof.)
Let $x_n\in X$ be a sequence which converges to $x\in\beta X\setminus X$. We can assume that $x_n$'s are distinct. I will show bellow that $\{x_n; n\in\mathbb N\}$ is closed discrete subspace of $X$. But first I will show how to use this fact.
For any choice of $y_n\in[0,1]$, $n\in\mathbb N$, we can define $f(x_n)=y_n$ and extend it continuously (by Tietze's theorem) to the whole $X$. Now there exists a continuous extension $\overline f : \beta X \to [0,1]$. By continuity, the sequence $y_n=\overline f(x_n)$ converges to $\overline f(x)$. We have shown that every sequence in $[0,1]$ is convergent, a contradiction.
Now to the proof that $\{x_n; n\in\mathbb N\}$ is closed and discrete.
Since $\{x_n; n\in\mathbb N\}\cup\{x\}$ is a compact subset of $\beta X$, it is closed in $\beta X$. The intersection with $X$ is $\{x_n; n\in\mathbb N\}$ and it must be closed in $X$.
Now we consider $\{x_n; n\in\mathbb N\}$ as a subspace of $X$ and we want show that this subspace is discrete. Choose some $x_n$. By Hausdorffness, it can be separated from $x$, i.e. there exists a neighborhood $U\ni x$ such that $x_n\notin U$ and a neighborhood $V\ni x_n$ with $V\cap U=\emptyset$.
Now by convergence $U$ contains all but finitely many $x_n$'s, hence using Hausdorfness we can separate $x_n$ from the (finitely many) remaining ones.
In Gillman & Jerison's Rings of continuous functions I found the following note
8.21 N.B. A number of authors have fallen into the trap of assuming then every countable, closed, discrete subset of a completely regular space is $C^\ast$-embedded. We have just seen a counterexample: [...]
It seems likely that one of these authors, or someone following them has made up this problem, as the same counterexample works here. (Martin Sleziak's answer shows how the assumption would lead to a proof).
The counterexample referred to is the Tychonoff plank $([0, \omega_1] \times [0, \omega]) \setminus (\omega_1, \omega)$. It can be shown that every real-valued function on that space can be extended to its one-point compactification $[0, \omega_1] \times [0, \omega]$. That implies that this is also the Stone-Čech compactification, and clearly $\lim_{n\to\infty} (\omega_1, n) = (\omega_1, \omega)$.