I need help in the following questions :

Qn1 : Let f ; g be continuous functions from [0, 1] onto [0, 1]. Prove that there is a ∈ [0, 1] such that f(g(a)) = g(f (a)).

Qn2 : Compute the limit lim(n→∞) integrate (0,pi,cos(x^n))

Thanks in advance.

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- Jan 27th 2010, 08:32 AM #1

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- Jan 2010
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## need help

I need help in the following questions :

Qn1 : Let f ; g be continuous functions from [0, 1] onto [0, 1]. Prove that there is a ∈ [0, 1] such that f(g(a)) = g(f (a)).

Qn2 : Compute the limit lim(n→∞) integrate (0,pi,cos(x^n))

Thanks in advance.

- Jan 27th 2010, 09:20 AM #2
(1)

First observe that and are onto since and are.

Suppose that for all . This means that for some . And for all implies there exists a such that .

These are contradictions, hence the conclusion follows. (By continuity)

(You can use intermediate value theorem to show: and for some implies there exists a such that , by continuity of ).