$ = \displaystyle\lim_{x \to +\infty} \frac{1}{2^x} \left( 2 + \frac{3}{x} \right )^x = $
$ = \displaystyle\lim_{x \to +\infty} \left(\frac{2}{2} + \frac{3}{2x} \right )^x = \displaystyle\lim_{x \to +\infty} \left(1 + \frac{3}{2x} \right )^x = e^{\frac{3}{2}} = \sqrt{e^3} $
Porti 2^(-x) dentro la parentesi e diventa
lim (1+3/(2x))^x
Cambio variabile : 1/t =3/(2x) , x = 3t/2
lim t -> +oo (1+1/t)^(3t/2)=limite notevole = e^(3/2)