Interesting Limit (4)

$\begin{aligned} L&=\lim\limits_{n\rightarrow\infty}\sum_{k=1}^n\sin\frac{2k-1}{n^2} \\&=\lim\limits_{n\rightarrow\infty}\frac1{2\sin\frac1{n^2}}\sum_{k=1}^n 2\sin\frac{2k-1}{n^2}\sin\frac1{n^2} \\&=\lim\limits_{n\rightarrow\infty}\frac1{2\sin\frac1{n^2}}\sum_{k=1}^n\left(\cos\frac{2k-2}{n^2}-\cos\frac{2k}{n^2}\right) \\&=\lim\limits_{n\rightarrow\infty}\frac1{2\sin\frac1{n^2}}\left(1-\cos\frac2n\right) \\&=\lim\limits_{n\rightarrow\infty}\frac{\sin^2\frac 1n}{\sin\frac1{n^2}} \\&=\lim\limits_{n\rightarrow\infty}\frac{\frac 1{n^2}}{\sin\frac1{n^2}}\cdot\left(\frac{\sin \frac 1n}{\frac 1n}\right)^2 \\&=\boxed 1 \end{aligned}$