Computing $\lim\limits_{n\to+\infty}n\int_{0}^{\pi/2}xf(x)\cos ^n xdx$
I got stuck at the following problem.
Let $f\in C([0,\pi/2])$, then compute $$ \lim_{n\to+\infty}n\int\limits_{0}^{\pi/2}xf(x)\cos ^n xdx $$
Could you suggest a helpful idea?
Here is a statistical solution. Let $X_1,\dots, X_n$ be i.i.d. random variables on $[0,\pi/2]$ with density $f_X(x)=\sin(x)$. The distribution function of $X$ is $F(x)=1-\cos(x)$. Now let $M=\min(X_1,\dots, X_n)$; its density function is $$f_M(x)=n(1-F(x))^{n-1}f_X(x)=n\,\cos^{n-1}(x)\sin(x).$$ Also, it is not hard to see that $M\to 0$ in distribution as $n\to\infty$. Now $$\int_0^{\pi/2} n \cos(x)^n xf(x)\,dx =\int_0^{\pi/2} f_M(x) \,\cos(x)\,{x\over \sin(x)}f(x)\,dx =\mathbb{E}\left(\cos(M)\,{M\over \sin(M)}\,f(M)\right).$$ Since $f$ is continuous, this converges to $\cos(0)\cdot1\cdot f(0)=f(0)$ as $n\to\infty$.
Note that $|x-\sin x|\ll x$ when $x\to0$. Hence, as soon as the function $f:[0,\pi/2]\to\mathbb R$ is
- measurable bounded, 2. continuous at $0$,
(no other property being necessary for the proof below to hold), there exists a function $g:[0,\pi/2]\to\mathbb R$ such that
- $g$ is measurable bounded, 2. $g(x)\to0$ when $x\to0$,
and such that, for every $x$ in $[0,\pi/2]$, $$ xf(x)=f(0)\sin x+g(x)\sin x. $$ Thus, the $n$th integral one is interested in is $$ I_n=\dfrac{n}{n+1}(f(0)\,J_n+K_n), $$ with $$ J_n=(n+1)\int_0^{\pi/2}\sin x\,(\cos x)^n\,\mathrm dx=\left[-(\cos x)^{n+1}\right]_0^{\pi/2}=1, $$ and $$ K_n=(n+1)\int_0^{\pi/2}g(x)\sin x\,(\cos x)^n\,\mathrm dx. $$ By (1.), there exists $C$ such that $|g(x)|\leqslant C$ for every $x$. By (2.), for every $\varepsilon\gt0$, there exists $x_\varepsilon\gt0$ such that $|g(x)|\leqslant\varepsilon$ for every $x\leqslant x_\varepsilon$. Hence, $$ |K_n|\leqslant\varepsilon J_n+(n+1)C\int_{x_\varepsilon}^{\pi/2}\sin x(\cos x)^n\,\mathrm dx=\varepsilon+C\,(\cos x_\varepsilon)^{n+1}. $$ When $n\to\infty$, $(\cos x_\varepsilon)^{n+1}\to0$ because $x_\varepsilon\gt0$, hence $\limsup\limits_{n\to\infty}|K_n|\leqslant\varepsilon$. This holds for every $\varepsilon\gt0$ hence $\lim\limits_{n\to\infty}K_n=0$. Finally, $$ \lim\limits_{n\to\infty}I_n=f(0). $$
This answer was completed using Robert's ideas in the comments.
By the Weierstraß approximation theorem, we can approximate $f$ arbitrarily well (in the supremum norm) by a polynomial. For a polynomial $p$, we have
$$ \begin{align} n\int_0^{\pi/2}xp(x)\cos ^n x\,\mathrm dx &= \int_0^{\pi/2}\frac{x\cos x}{\sin x}p(x)n\sin x\cos ^{n-1} x\,\mathrm dx \\ &= \left[-\frac{x\cos x}{\sin x}p(x)\cos^n x\right]_0^{\pi/2}+\int_0^{\pi/2}\left(\frac{x\cos x}{\sin x}p(x)\right)'\cos^n x\,\mathrm dx\;. \end{align} $$
The boundary term evaluates to $p(0)$ for all $n$, and the integral goes to $0$, so the limit is $p(0)$. Now
$$ \begin{align} \left|n\int_0^{\pi/2}xf(x)\cos ^n x\,\mathrm dx-n\int_0^{\pi/2}xp(x)\cos ^n x\,\mathrm dx\right| &= n\int_0^{\pi/2}x|f(x)-p(x)|\cos ^n x\,\mathrm dx \\ &\le \epsilon n\int_0^{\pi/2}x\cos ^n x\,\mathrm dx\;, \end{align} $$
which is just $\epsilon$ times the desired integral for $f\equiv1$ and thus by the above goes to $\epsilon$ for $n\to\infty$. Since both the function values at $0$ and the limits of the integrals differ at most by $\epsilon$, which can be made arbitrarily small, the desired limit is $f(0)$.