Limit of a sequence

Let f : [0,1] --> [0,1] be continuous and strictly decreasing such that
f(0)=1, f(1)=0.
Is it true that

lim_(n-->oo) [ int (f(x))^(n+1) dx ] / [ int (f(x))^n dx ] = 1 (?)

If the limit in (?) exists, then it must be 1 ( = ||f||_oo ),
but I think that the sequence diverges for some f.
I could not find such an f. Can you help?

Thank you.

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