Problem of today's MATH202 tutorial lesson

Too many hints are present in this note.

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In example 4 we have come across a boring question. Here is a much interesting one.
$latex \boxed{\text{\textbf{Problem}}}$
Suppose $latex f''(0)$ exists and $latex f''(0)\neq 0$. Prove that in the mean value theorem $latex f(x)-f(0)=xf'(\theta x)$, we have $latex \displaystyle \lim_{x\to 0}\theta =\frac{1}{2}$.

This is a problem of my past assignment in MATH203, and the example 4 is exactly one of them.