Does the following $limit$ exist? $If$ yes, find $it.$

$(a)If$ you are not $in$ honors program, $do$ this problem:

$\underset{n}{lim}\frac{nsin(n!)}{n^2+1}n!$ denotes $n$ factorial.$\underset{n}{lim}{x}_n,$

where

$x_n=\sqrt[\phantom{2}]{7+\sqrt[\phantom{2}]{7+\text{d}\text{o}\text{t}\text{s}+\sqrt[\phantom{2}]{7}}}$

and $in$ that expression $n$ roots are taken.