Question on limits and integration

by aditya bhatt   Last Updated October 20, 2019 05:20 AM - source

Let $f$ be a continuous function on $[0,1]$. Then the $\lim_{n\to ∞}\int_{0}^{1}nx^nf(x)$ is equal to ?

What I did: I tried for $f(x)=x$ and on computing the integral and the limit I got the answer = $1 \implies f(1)$[ according to the options]. However, I do not know if this is a valid way to generalize the answer for all functions continuous on $[0,1]$.

Can anyone please help me solve it?

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