1. ## Limit (3)

For any positive integers $\displaystyle m,n$ define $\displaystyle f(m,n)=\int_{\frac{1}{m}}^{\frac{\pi}{2}} \frac{\sin(nx)}{\sin x} \ dx.$ Evaluate $\displaystyle L_1=\lim_{m\to\infty} \lim_{n\to\infty} f(m,n)$ and $\displaystyle L_2=\lim_{n\to\infty} \lim_{m\to\infty} f(m,n).$

2. The function $\displaystyle \frac1{\sin x}$ is in $\displaystyle \textrm C[{\textstyle\frac1m},{\textstyle\frac\pi2}]\subset\mathrm L^1[{\textstyle\frac1m},{\textstyle\frac\pi2}]$, and by the Riemann-Lebesgue lemma $\displaystyle \lim_{n\to\infty}\int_{1/m}^{\pi/2}\frac{\sin nx}{\sin x}\mathrm dx=0$. Therefore $\displaystyle \lim_{m\to\infty}\lim_{n\to\infty}f(m,n)=0$.

The function $\displaystyle \frac{\sin nx}{\sin x}$ is in $\displaystyle \mathrm C[0,{\textstyle\frac\pi2}]$ (with $\displaystyle f(0)=n$) and is improperly Riemann integrable there, so $\displaystyle \lim_{m\to\infty}\int_{1/m}^{\pi/2}\frac{\sin nx}{\sin x}\mathrm dx=\int_0^{\pi/2}\frac{\sin nx}{\sin x}\mathrm dx$.

The function $\displaystyle \frac1x-\frac1{\sin x}$ (with $\displaystyle f(0)=0$) is in $\displaystyle \mathrm C[0,{\textstyle\frac\pi2}]$, and is improperly Riemann integrable and Lesbesgue integrable there.

By the Riemann-Lebesgue lemma, $\displaystyle \lim_{n\to\infty}\int_0^{\pi/2}\left(\frac{\sin nx}{x}-\frac{\sin nx}{\sin x}\right)\mathrm dx=0$.

Thus $\displaystyle \lim_{n\to\infty}\int_0^{\pi/2}\frac{\sin nx}{\sin x}\mathrm dx=\lim_{n\to\infty}\int_0^{\pi/2}\frac{\sin nx}x\mathrm dx=\lim_{n\to\infty}\int_0^{n\pi/2}\frac{\sin x}x\mathrm dx=\int_0^\infty\frac{\sin x}x\mathrm dx=\frac\pi2$. Therefore $\displaystyle \lim_{n\to\infty}\lim_{m\to\infty}f(m,n)=\frac\pi2$.

3. Originally Posted by halbard
The function $\displaystyle \frac1{\sin x}$ is in $\displaystyle \textrm C[{\textstyle\frac1m},{\textstyle\frac\pi2}]\subset\mathrm L^1[{\textstyle\frac1m},{\textstyle\frac\pi2}]$, and by the Riemann-Lebesgue lemma $\displaystyle \lim_{n\to\infty}\int_{1/m}^{\pi/2}\frac{\sin nx}{\sin x}\mathrm dx=0$. Therefore $\displaystyle \lim_{m\to\infty}\lim_{n\to\infty}f(m,n)=0$.

The function $\displaystyle \frac{\sin nx}{\sin x}$ is in $\displaystyle \mathrm C[0,{\textstyle\frac\pi2}]$ (with $\displaystyle f(0)=n$) and is improperly Riemann integrable there, so $\displaystyle \lim_{m\to\infty}\int_{1/m}^{\pi/2}\frac{\sin nx}{\sin x}\mathrm dx=\int_0^{\pi/2}\frac{\sin nx}{\sin x}\mathrm dx$.

The function $\displaystyle \frac1x-\frac1{\sin x}$ (with $\displaystyle f(0)=0$) is in $\displaystyle \mathrm C[0,{\textstyle\frac\pi2}]$, and is improperly Riemann integrable and Lesbesgue integrable there.

By the Riemann-Lebesgue lemma, $\displaystyle \lim_{n\to\infty}\int_0^{\pi/2}\left(\frac{\sin nx}{x}-\frac{\sin nx}{\sin x}\right)\mathrm dx=0$.

Thus $\displaystyle \lim_{n\to\infty}\int_0^{\pi/2}\frac{\sin nx}{\sin x}\mathrm dx=\lim_{n\to\infty}\int_0^{\pi/2}\frac{\sin nx}x\mathrm dx=\lim_{n\to\infty}\int_0^{n\pi/2}\frac{\sin x}x\mathrm dx=\int_0^\infty\frac{\sin x}x\mathrm dx=\frac\pi2$. Therefore $\displaystyle \lim_{n\to\infty}\lim_{m\to\infty}f(m,n)=\frac\pi2$.
beautiful! my solution is the same as yours except that i proved $\displaystyle \lim_{n\to\infty}\int_{1/m}^{\pi/2}\frac{\sin nx}{\sin x}\mathrm dx=0$ and $\displaystyle \lim_{n\to\infty}\int_0^{\pi/2}\left(\frac{\sin nx}{x}-\frac{\sin nx}{\sin x}\right)\mathrm dx=0$ without using the Riemann-Lebesgue lemma.

so i'll leave that open to whoever wants to try it.