(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let A, B be subsets of ℝ and f : A → ℝ and g : B → ℝ and f(x)[itex]\in[/itex]B for every x [itex]\in[/itex] A.

Prove: If f(x)→L as x → a, for x[itex]\in[/itex] I and g is continuous at L [itex]\in[/itex] B then

lim x->a g(f(x)) = g(lim x->a f(x)).

2. Relevant equations

f is said to be continuous at a point a iff given ε>0 there is a δ>0. such that

|x-a|<δ implies |f(x) - f(a)|<ε.

3. The attempt at a solution

By the premise:

Exists a [itex]\delta_{1}[/itex] such that |x-a|< [itex]\delta_{1}[/itex] implies |f(x)-L|< [itex]\delta_{2}[/itex].

So choose an x s.t. |x-a|< [itex]\delta_{1}[/itex].

Since g is continuous:

Exists a [itex]\delta_{2}[/itex] such that |f(x)-L|<[itex]\delta_{2}[/itex] implies

|g(f(x)) - g(L)| <ε.

Is this correct?

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# Homework Help: Limits of composition of two functions.

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