$$ \text{Let f(x) =} \lim_{x \to \infty} \left( \frac{n^n(x+n)(x+2n) \cdot \cdot \cdot (x+\frac{n}{n})}{n!(x^2+n^2)(x^2+\frac{n^2}{4}) \cdot \cdot \cdot (x^2+\frac{n^2}{n^2})} \right)^{\frac{x}{n}} \forall\text{ x > 0. Then:} \
(a)f \left( \frac{1}{2}\right)\ge f(1) \qquad \qquad\qquad\qquad (b)f \left( \frac{1}{3}\right)\le f(\frac{2}{3}) \
(c)f' \left( 2\right)\le 0\qquad \qquad\qquad\qquad \qquad(d)\frac{f'(3)}{f(3)} \ge \frac{f'(2)}{f(2)} $$
#An interesting Hard Question for High Schoolers
loud swift
