Homework Help: Limits of composition of two functions.

1. Dec 12, 2012

Max.Planck

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)$\in$B for every x $\in$ A.
Prove: If f(x)→L as x → a, for x$\in$ I and g is continuous at L $\in$ 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 $\delta_{1}$ such that |x-a|< $\delta_{1}$ implies |f(x)-L|< $\delta_{2}$.
So choose an x s.t. |x-a|< $\delta_{1}$.
Since g is continuous:
Exists a $\delta_{2}$ such that |f(x)-L|<$\delta_{2}$ implies
|g(f(x)) - g(L)| <ε.

Is this correct?

Last edited: Dec 12, 2012