I substituted $n=\frac 1{x^2}$, which also means that $x^2=\frac 1 n$.
So we get $\Big(1 + \frac 32 x^2\Big)^{1/x^2} = \Big(1 + \frac 32 \cdot \frac...
Not quite, although you started correctly. The limit in this case is as $x\to\infty$, so you want to see what happens when $x$ gets large. This means...
Ok! That means that $f$ is monotone and escpecially descreasing, right? To get that $f$ is strictly monotone, do we have to get $f'(x)<0$ instead of...