Uniform convergence of $f_n = \frac{nx}{1+nx^2}$.
Hint 1: For $|x|\ge M$, $$ \begin{align} \left|\frac{nx}{1+nx^2}-\frac1x\right| &=\frac1{x(1+nx^2)}\\ &\le\frac1{nM^3} \end{align} $$ Hint 2: The Uniform Limit Theorem.
Uniform convergence of $f_n = \frac{nx}{1+nx^2}$.
Hint 1: For $|x|\ge M$, $$ \begin{align} \left|\frac{nx}{1+nx^2}-\frac1x\right| &=\frac1{x(1+nx^2)}\\ &\le\frac1{nM^3} \end{align} $$ Hint 2: The Uniform Limit Theorem.