Thorem: If f(x) is continuous at L and lim_(x->a) g(x)=L, then lim_(x->a) f(g(x)=f(lim_(x->a) g(x))=f(L). Proof: Assume f(x) is continuous at a point L, and that lim_(x->a) g(x)=L.

Modelfino0g 2022-09-14 Answered
Thorem: If f ( x ) is continuous at L and lim x a g ( x ) = L, then lim x a f ( g ( x ) = f ( lim x a g ( x ) ) = f ( L ).
Proof: Assume f ( x ) is continuous at a point L, and that lim x a g ( x ) = L.
ϵ > 0 , δ > 0 : [ | x L | < δ | f ( x ) f ( L ) | < ε ].
And δ > 0 , δ > 0 : [ | x a | < δ | g ( x ) L < δ ].
So, δ > 0 , δ > 0 : [ | x a | < δ | f ( g ( x ) ) f ( L ) | < ϵ ].
lim x a g ( x ) = L so f ( lim x a g ( x ) ) = f ( L ).
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Answers (1)

wegpluktee3
Answered 2022-09-15 Author has 12 answers
You're doing the right thing but the way you've presented it is a bit confusing. Why not word it like this:
Pick ϵ > 0. Continuity of f at L gives you an η such that | x L | < η | f ( x ) f ( L ) | < ϵ.
For this η, lim x a g ( x ) = L gives you a δ such that | x a | < δ | g ( x ) L | < η.
Hence | x a | < δ | g ( x ) L | < η | f ( g ( x ) ) f ( L ) | < ϵ

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