$$\left(1-\frac1{n^2}\right)^n = \left(1-\frac1{n^2}\right)^{(n^2\frac{n}{n^2})} = \left(\left(1-\frac1{n^2}\right)^{n^2}\right)^{\frac1n}$$
The expression inside the (outer) brackets is easily transformed into a standard result.
Similar manipulations deal with the other examples. For the second, write $$n^2+1=(n^2-1)+2$$