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Limits of composition of two functions.

  1. Dec 12, 2012 #1
    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?
     
    Last edited: Dec 12, 2012
  2. jcsd
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