I have this sequence of functions $f_{n}(x)=\frac{n^{\frac{3}{2}}x}{1+n^{2}x^{2}}$ for all $x \in [0,1]$
I need to show that $f_{n}(x)$ goes to 0 and it's not uniformly bounded? And if we have a Lebesgue integrable function $h(x)=\frac{1}{x^{\frac{1}{2}}}$ on $[0,1]$ can we show that $f_{n}<g$ for all n and $x\in (0,1]$???
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Put $x_n = 1/n$. Then $f_n(x_n) = \sqrt{n}/2$ so your sequence of functions is not uniformly bounded. As for why the sequence goes to the zero function, just take the limit as $n \to \infty$ and you will get that the denominator terms dominates in asymptotic growth, unless $x = 0$ but your function is automatically $0$ at $x = 0$.
Take $x=1/n$ and let $n\to\infty$.