$f$ be a function defined on some subset $D$. The function has a limit $l$ at $0$ iff for every sequence $(x_n)$ in $D$ converging to $0$, the sequence $f(x_n)$ converges to $l$.
Now, if for any sequence of rational numbers $(a_n)$ and any sequence of irrational numbers $(b_n)$ converging to $0$, if $f(a_n)$ and $f(b_n)$ converge to $l$, does that imply the limit at $0$ is $l$? I know that I left out those sequences which have both rational and irrational elements. Just wondeing if the limit exists in such a case. Thanks in advance!