I want to determine if the sequence of functions $f_{n}(x)=\sqrt{nx}\arctan\left(\frac{1}{nx}\right)$ converges uniformly to its pointwise limit on $x>0$. To determine the limit, we can Taylor expand the arctan to find that as $\lim_{n\to\infty}f_n(x)=0$, which is correct according to the answer in the text book.
To determine if the convergence is uniform, I want to look at $\lim_{n\to\infty}\sup_{x>0}|f_n(x)|$. I started by attempting to take the derivative and find the maxima, but the equation is transcendental, so determining a maximum becomes quite difficult. According to the answer in the book, the convergence is not uniform.
Any help to show this is the case would be greatly appreciated!