Geodesic
Limits of composite functions
The Teaching of Mathematics, XII (2009) no. 1, p. 1

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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.
Classification : 1AMS00A35 2ZDMI25
Keywords: Composite function, introduction of a new variable. 
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Milosav M. Marjanović; Zoran Kadelburg. Limits of composite functions. The Teaching of Mathematics, XII (2009) no. 1, p. 1 . http://geodesic.mathdoc.fr/item/TM2_2009_XII_1_a0/