$\underset{t\to 0}{\lim }{(1^{\frac{1}{{ }^{{\sin }^{2}t}}}+2^{\frac{1}{{ }^{{\sin }^{2}t}}}+\mathrm{...............}+n^{\frac{1}{{ }^{{\sin }^{2}t}}})}^{{ }^{{\sin }^{2}t}}$ is equal to 

$\underset{t\to 0}{lt}{(1^{\cos e{c}^{2}t}+{2^{\cos ec}}^{{ }^{2}t}+\mathrm{.....}+n^{\cos e{c}^{2}t})}^{{\sin }^{2}t}$

$\underset{t\to 0}{lt}n{[{\left(\frac{1}{n}\right)}^{\cos e{c}^{2}t}+{\left(\frac{2}{n}\right)}^{\cos e{c}^{2}t}+\mathrm{.......}+1]}^{{\sin }^{2}t}=n(0+0+\mathrm{.....}+1)=n$