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![[parent]](http://images.planetmath.org:8080/images/uparrow.png) Viewing Message
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| ``Re: limit rule of compound function''
by azdbacks4234 on 2008-01-27 11:13:02 |
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| It's possible I wasn't quite precise enough. Consider the following: let S, T be subsets of \mathbb{R}, f:S\rightarrow\mathbb{R}, g:T\rightarrow \mathbb{R}R, and f(S)\subset T. Fix a\in S, and assume \lim_{x\rightarrow a}f(x) exists and equals b. Assume further that \lim_{y\rightarrow b}g(y) exists and equals c. Then
\lim_{x\rightarrow a}g(f(x))=c.
Given \epsilon>0, there exists \delta_1>0 such that if y\in T and 0<|y-b|<\delta_1, then |g(y)-b|<\epsilon. Select \delta_2>0 such that for all x\in S with 0<|x-a|<\delta_1, |f(x)-b|<\delta_1. It follows that for all such x, |g(f(x))-b|<\epsilon, so that \lim_{x\rightarrow a}g(f(x))=b. |
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