Can’t figure out whether a sequence of functions converges uniformly.

Question

The sequence in question is $ f_n(x)= 1 - \exp (- \frac{nx^2}{nx + 1})$, and the interval is [ 1/2 , 3 ] I tried finding the sup of the difference between this and the limit function, using the derivative, to no avail.

Answer 1

Note that $$\frac{(n+1)x^2}{(n+1)x+1} = 1+ \frac{1}{n^2 x+n x+n}\uparrow1,$$ from which we conclude that the sequence $f_n$ is monotone ($f_{n+1}\geqslant f_n$ for all $n$), with the pointwise limit $f(x):=1-e^{-x}$.

As $f_n$ is defined on the compact space $[1/2, 3]$ and $f$ is continuous, by Dini's theorem it follows that the convergence is uniform.