Integral Calculus: JEE Advanced 2023 Mathematics question

Let $f:(0,1) \rightarrow \mathbb{R}$ 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 $g:(0,1) \rightarrow \mathbb{R}$ be a function such that $\int_{x^{2}}^{x} \sqrt{\frac{1-t}{t}} d t<g(x)<2 \sqrt{x}$ for all $x \in(0,1)$. Then $\lim _{x \rightarrow 0} f(x) g(x)$