Supremum and Limit superior

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Let $g_n$ be a sequence of functions, f a function.

Why, in general, is $$\sup_n(g_n\circ f)\ne\sup_n(g_n)\circ f$$ while $$\limsup_n(g_n\circ f)=\limsup_n(g_n)\circ f$$ holds?