As a followup question to this question: The relation between energy quanta of an Einstein solid and the equipartition value of heat capacity and this answer https://physics.stackexchange.com/a/133053/55779:
Apparently energy quanta and the equipartition values are related somehow.
I'd like to approach the problem without the formula, since the problem the question is asked in is before the problem where that formula is derived.
How can I know that value of $x\approx3$ without the formula ? I can't even tell that I should be looking for the half maximum. Or can I?