New Answer: It indeed follows from vertex-multiplication. If you replace each edge with a stable set of size $k$, then $\alpha(G)/n$ will not change, while $n^r$ and the number of edges both grow by a $k^r$ factor. This shows that $\lim_{n \to \infty}\frac{RT(n,L^{(r)},\varepsilon n)}{n^r}$ has a limit by a [Fekete lemma][1] type argument.

Old Answer: Many of these functions are not known to have a limit. In this case when someone proves a bound on them, in fact they show a bound for the liminf or limsup. Unfortunately, in many papers they refer to it as lim, which also bothers me a lot. Incidentally, I've just asked the same thing two weeks ago (about the uniform Turan density function) from Danial Kral'.


  [1]: http://$%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%5Cfrac%7BRT(n,L%5E%7B(r)%7D,%5Cvarepsilon%20n)%7D%7Bn%5Er%7D$