I need help with the following theorem:(adsbygoogle = window.adsbygoogle || []).push({});

Let I, J ⊆ℝ be open intervals, let x∈I, let g: I\{x}→ℝ and f: J→ℝ be functions with g[I\{x}]⊆J and Lim_{z→x}g(x)=L∈J. Assume that lim_{y→L }f(y) exists and that g[I\{x}]⊆J\{g(x)},or, in case g(x)∈g[I\{x}] that lim_{y→L}f(y)=f(L). Then f(g(x)) converges at x, and lim_{z→x}f(g(x))=lim_{y→L}f(y).

I don't get why the assumptions are necessary. We're assuming that the limit of f exists when g is not defined at x, or, if g is defined at x, that the lim_{y→L}f (y)=f(L).

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# Limit of Composite Function

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