The sequence is: $f_n(x) = \dfrac{nx^2+1}{2n+x}$ with derivative $f_n'(x) = \dfrac{4n^2x+nx^2-1}{4n^2+4nx+x^2}$. We know that $f'(x) = x$.
We are asked to show that the sequence of derivatives, $f_n'(x) = \dfrac{4n^2x+nx^2-1}{4n^2+4nx+x^2}$, converges uniformly every interval [-M,M].
I've been struggling with this problem. Using the definition and some manipulation, I get that $|f_n'(x)-f'(x)| = |\dfrac{x^3+3nx^2+1}{x^2+4xn+4n^2}|$, but don't know where to go from here. How can I get this last part bounded? Would appreciate any help on this! Thank you!