Consider the function $f$ which is continuous. Calculate $\lim_{n \to \infty} \int_0^1nx^nf(x)dx$.
Here first I attempted to prove $f_n=nx^n$ is uniformly convergent using sup-norm limit but unfortunately, it is not uniformly convergent, as $M_n=sup_{x\in[0,1]}f_n(x)=1$ so $M_n \not\to 0$. So, I guess that the limit would not be zero.
I have an intuition that it would be $1$, but can't prove it. Any help!!