How to infer if $\frac{n x^s}{1+nx}$ increasing?

Question

How to infer IF $\frac{n x^s}{1+nx}$ increasing? It necessarily isn't, but I speculated it could be.

$0<s<1$, $n \rightarrow \infty$, $x \in [0,1]$.

I tried the derivative $f_n'(x)$ but it didn't seem conclusive.

Answer 1

$$ f'_n(x)=\frac{n x^{s-1 } (s - n x + n s x)}{(1 + n x)^2} $$ Assuming $n$ is large enough so that $n(1-s)>1$ the derivative is positive when: $$ 0\le x< \frac{s}{n(1-s)} <1 $$ When $n$ increases this interval decreases. For relatively small $n$ such that $n(1-s)<1$ : $$ 1\ge x>\frac{s}{n(1-s)}\gt0 $$

Note that as $ n\to\infty$, $\frac{s}{n(1-s)}$ becomes smaller and smaller.

Answer 2

As $n\to\infty,f_n(x)=\frac{x^s}{x+1/n}$ behaves like $f(x)=x^{s-1}~\forall x>0$. Since $s<1,f(x)$ is strictly decreasing for $x>0$.