Here's a fun integral question I made for a competition a few years ago. I was inspired by this much easier question:
Warmup: Let f(x) be a continuous real-valued function on [0,1]. Calculate: \[\lim_{n\to\infty}\int_0^1x^nf(x)dx\]
Now for the real question: Let f(x) be a continuous real-valued function on [0,1]. Calculate \[\lim_{n\to\infty} n\int_0^1x^nf(x)dx\]

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The first one is clearly zero.
The second one .... one sec

f(1)

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