Let $\displaystyle f$ and $\displaystyle g$ be functions with continuous second derivatives on $\displaystyle [0,1]$ and $\displaystyle f^{'}(0)g^{''}(0)-f^{''}(0)g^{'}(0) \neq 0$. Define a function $\displaystyle \theta$ for $\displaystyle x \in (0,1)$ so that $\displaystyle \theta (x)$ is one of the values that satisfies the generalised mean value theorem,

$\displaystyle \frac{f(x)-f(0)}{g(x)-g(0)} = \frac{f^{'}(\theta (x))}{g^{'}(\theta (x))}$.

Show that

$\displaystyle \lim_{x \to 0^+} \frac{\theta (x)}{x} = \frac{1}{2}$.