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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 perucho on 2008-01-27 01:26:07 |
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| Continuity of g at x=a is essential.
A\subset{R}---f--->B\subset{R}---g--->R(the reals)
x---------->y=f(x)---------->z=g(y)
A\subset{R}--------gof---------->R
x--------------------------->(gof)(x):=g(f(x))
codomain{f} \subset domain{g} \implies gof is defined.
\lim_{x\to x_0}f(x)=a (possibly not equals to f(x_0))
g *continuos* at y=a \implies \lim_{y\to a}g(y)=g(a),
so g(\lim_{x\to x_0}f(x))=\lim_{f(x)\to a} g(f(x)) =\lim_{x\to x_0}g(f(x)),
since as f(x)\to a then x\to x_0, as limit of f(x) exists at x=x_0 by hypothesis. |
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