For each of the following sequences of functions on $[0,1]$, indexed by $n=1,2, \ldots$, determine whether or not the sequence has a pointwise limit, and if so, determine whether or not the convergence to the pointwise limit is uniform.

1. $f_{n}(x)=1 /\left(1+n^{2} x^{2}\right)$

2. $g_{n}(x)=n x(1-x)^{n}$

3. $h_{n}(x)=\sqrt{n} x(1-x)^{n}$