The general thing to keep in mind with these problems is to just build your example a step at a time, keeping everything as simple as possible.
First we need a convergent sequence $x_n$. The simplest possible convergent sequence is a constant sequence, but that can't work here. The next simplest is probably $x_n=1/n$, converging to $0$.
Now we need $f(x_n)$ to converge. Again, the simplest way for this to happen is for $f(x_n)$ to be constant, so let's set $f(1/n)=0$ for all $n$.
Finally, we need $f(l)$ to not be the limit of $f(x_n)$. In other words, $f(0)$ has to not equal $0$. This is easily accomplished, just set $f(0)=1$ say.
All that remains is to define $f$ for all other points. Since $f$ already satisfies all the conditions, we can do this however we like. For the sake of argument, we could set $f(x)=0$ wherever $f(x)$ wasn't already defined.