Uniform convergence of $\lim_{n \to \infty} \frac{nx}{1 + n^2x^2} \text{ where } 0 \le x \le 1$

Question

Prepering to second exam in Calculus 2,

and I remember this question from the first exam that I fell on:

Let $f_n(x)$ be the function sequence $\{\frac{nx}{1 + n^2x^2}\}_{n=1}^\infty$

Let $f(x)$ be the limit function such that $\lim_{n \to \infty} \frac{nx}{1 + n^2x^2}$ convergences to.

Does $f_n(x)$ convergents uniformly to $f(x)$?

As for the answers, the answer is no.

Can you please explain me why?

Thanks in advance!

Answer 1

If you investigate the $$c_n=\max\mid f_n(x)-f(x)\mid=\max\limits_{x\in[0,1]}\mid \frac{nx}{1+n^2x^2}\mid$$ then you find that it takes its maximum at $x=\frac{1}{n}$ so $$c_n=\frac{n.\frac{1}{n}}{1+n^2\left(\frac{1}{n}\right)^2}=\frac{1}{2} $$which means that $f_n(x)$ converges nonuniformly.