In the book mathematical analysis by S K Chatterjea (2nd edition) this limit is mentioned on page 72. 
For integers $x$, $$kx - [kx] = 0 $$ hence $$l = 0$$.
For irrationals, if we except the difference (i.e, $kx - [kx]$) to be random. then the average value would be 0.5 as the difference lies between 0 and 1. thus $l = 0.5$.
for rationals, we could assume it to random but the difference is 0 every time $q$ divides $k$ , hence its average value would be less than 0.5
This sounds plausible but how we prove it?
I couldn't locate any sources online. i would be grateful to if someone could point me to any references. Does this limit have some name and has been studied in literature?