How to find the limit correctly?

I want to use MMA to find the limit of $lim _{n rightarrow infty}left(frac{sin frac{pi}{n}}{n+1}+frac{sin frac{2 pi}{n}}{n+frac{1}{2}}+cdots+frac{sin pi}{n+frac{1}{n}}right)$.

Limit[Sum[Sin[(i*Pi)/n]/(n + 1/i), {i, 1, n}], n -> Infinity]

However, the above code does not get the correct result (the answer is $frac{2}{pi}$).

What can I do to get the right results?

AsymptoticSum[Sin[(i*Pi)/n]/(n + 1/i), {i, 1, n}, {n, Infinity, 1}]
(*2/[Pi]*)