Let \( f : (0,1) \to \mathbb{R} \) be the function defined as \( f(x) = \sqrt{n} \) if \( x \in \left[ \frac{1}{n+1}, \frac{1}{n} \right) \) where \( n \in \mathbb{N} \). Let \( g : (0,1) \to \mathbb{R} \) be a function such that \( \int_{x^2}^{x} \sqrt{\frac{1-t}{t}} \, dt < g(x) < 2\sqrt{x} \) for all \( x \in (0,1) \). Then \( \displaystyle \lim_{x \to 0} f(x) g(x) \)
