
This is not true that Q does not have limit points (if you are in topological spaces).
Here is my answer.
An element p of R is called limit point of Q if every open set G containing p contains the point of Q different from p. Set of all limit points is called derived set.
Now open sets in R are open intervals and union of open intervals. We can give the answer just by looking to open interval.
Now let p an element of R. If we consider any open interval which contains p then surely its intersection with Q is nonempty (you can think about it) because in any interval you can find at least two rational number (no doubt there are infinitely many).
In this manner every real number is limit point of Q and hence derive set of Q is R. 
By definition every point of real line is a limit point of set of rational number.For every point in the real line you can always construct a sequence in rational number which will converge to that point.So derived set will be full of R.
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