A problem concerning limits (Q1191610)

Let \(f: ]0,\infty[\to X\), where \(X\) is a topological space. Suppose that either the condition (1) \(\lim_{n\to\infty} f(nt)= \theta\), \(t\in H\), or the condition (2) \(\lim_{n\to\infty} f(nt)= \theta(t)\), \(t\in H\), holds for some open interval \(H\subset ]0,\infty[\). The problem considered in this paper is connected with the existence of \(\lim_{t\to\infty} f(t)\) under some additional assumptions on \(X\) and \(f\).