Limits of composite functions

Milosav M. Marjanović, Zoran Kadelburg

Finding $\lim\limits_{x_0}g(f(x))$, first the following two limits $$ (i)\quad łim_{x_0} f(x)=y_0 \qquad (ii)\quad łim_{y_0} g(y)=lpha $$ are found and then, it is taken that $\lim\limits_{x_0} g(f(x))=\alpha$. The existence of the limits under (i) and (ii) is the basis for this method, which is not legitimate in general. In this notice we give necessary and sufficient conditions for the legitimacy of this method.