Let $f : [0,1] \longrightarrow \mathbb{R}$ be a continuous function. Determine (with appropriate justification) the following limit: $$\lim_{n \longrightarrow \infty} \int_0^1 nx^n f(x)\,\mathrm{d}x$$
Let $f : [0,1] \longrightarrow \mathbb{R}$ be a continuous function. Determine (with appropriate justification) the following limit:
$$\lim_{n \longrightarrow \infty} \int_0^1 nx^n f(x)\,\mathrm{d}x$$