Pointwise convergence. As n increases, what happens? More and more points are included as not very far from f(x) = 0. In fact, there isn't a point that won't be included eventually.

Uniform Convergence. The problem is over on the left, near x = 0. A little elementary calculus shows the first derivative having a zero at x = 1/(1+n). This certainly moves toward x = 0. What is the value of the function at x = 1/(1+n)? Whatever it is, it's limit needs to be zero if we are to establish Uniform Convergence

It is .

Use your favorite logarithm technique to prove to what this converges.

Rewriting as may not be obvious, but this invites l'Hospital.

A single application (and a little algebra) produces -(1 + (1/n)). Well, that's clear enough. So, the limit of the logarithm is -1. Where does that leave us?