limit

[PURE MATH]

Express the following the limit values in terms of a definite integral and find them
$(1) I=\lim_{n \to \infty} \frac{1}{n}In(1+\frac{1}{n} )(1+\frac{2}{n} )...(1+\frac{n}{n} ) $
$(2) J=\lim_{n \to \infty}\frac{1}{n^2}(\sqrt{n^2-1}+\sqrt{n^2-2^2}+...+\sqrt{n^2-n^2} ) $
$(3) K=\lim_{n \to \infty}\frac{1}{n^3}(\sqrt{n^2+1}+2\sqrt{n^2+2^2}+...+n\sqrt{n^2+n^2} ) $

For positive integers $n$ , let
$S_n=\frac{1}{\sqrt{1} }+\frac{1}{\sqrt{2} } +...+\frac{1}{\sqrt{n} } $ $,$ $T_n=\frac{1}{\sqrt{1+\frac{1}{2} } }+\frac{1}{\sqrt{2+\frac{1}{2} } }+...+\frac{1}{\sqrt{n+\frac{1}{2} } } $
Find $\lim_{x \to \infty} \frac{T_n}{S_n} $