An example in my textbook is as follows. Determine the limit $f(x)$ of the sequence $f_n(x)$ where $$f_n(x)=\frac{nx+n^2x^2}{1+n^2x^2},\quad n=1,2,3,...$$ Also determine if the convergence $f_n \longrightarrow f$ is uniform.
The solution states that it is clear that $f_n(x) \longrightarrow x$ as $n\longrightarrow \infty$. However by my calculations (and wolframalpha) $\lim_{n\to\infty} f_n(x)=1$??? What am I doing wrong?
The solution then continues with $$f_n(x)-f(x)=\frac{nx+n^2x^2}{1+n^2x^2}-x={\frac{(n-1)x}{1+n^2x^2}}.$$
Once again by my calculations $$f_n(x)-f(x)=\frac{nx+n^2x^2}{1+n^2x^2}-x=\frac{nx+n^2x^2-x-n^2x^3}{1+n^2x^2}.$$ I have no idea how the final equality simplifies to $\frac{(n-1)x}{1+n^2x^2}$? Any help gratefully accepted.