Hi Guys was attempting this question and was wondering if I was doing the question correctly?
Determine whether or not the sequence of functions is uniformly convergent:-
$$h_n:[0,3]\to \mathbb{R}$$ $$h_n(x) = \frac{n^2x^3}{n^3+x}, x\in[0,3]$$
Checking point wise convergence first
$$\lim_{n\to \infty}h_n(x) = \lim_{n\to \infty}\frac{n^2x^3}{n^3+x}$$
Applying L Hopital's Rule
$$\lim_{n\to \infty}h_n(x) = x^3 \lim_{n\to \infty}\frac{2n}{3n^2} = x^3 \lim_{n\to \infty}\frac{2}{3n} = x^3 \frac{2}{3} \lim_{n\to \infty}\frac{1}{n} $$
$$x^3 \frac{2}{3} \lim_{n\to \infty}\frac{1}{n} = 0$$
$$Therefore \lim_{n\to \infty}h_n(x) = h(x) = 0$$
Checking for uniformly convergent
$$M_n = sup|h_n(x)-h(x)|,x\in [0,3]$$
$$|h_n(x)-h(x)|$$
$$\frac{n^2x^3}{n^3+x}$$
is it accurate to say the following when checking to see uniform convergence
$$\frac{n^2x^3}{(n^3+x)} < \frac{n^2}{n^3} <\frac{1}{n} = 0 $$
$$\lim{n \to \infty} $$
hence the sequence of functions is indeed uniformly convergent ?