극한 $\displaystyle \lim _ {m, n \rightarrow \infty} {\sum _ {k=1} ^ {n} \sum _ {j=1} ^ {m} \frac{\pi k} {n} \sin \left (\frac{\pi jk} {4mn} \right) \frac{1} {mn}}$ 의 값은?

• ①$\displaystyle 2 \left (1- \frac{2 \sqrt 2} {\pi} \right)$
• ②$\displaystyle 4 \left (1- \frac{\sqrt {2}} {\pi} \right)$
• ③$\displaystyle 4 \left (1- \frac{2 \sqrt {2}} {\pi} \right)$
• ④$\displaystyle 2 \left (1- \frac{\sqrt {2}} {\pi} \right)$