What is the limit as $x \to 0$ of the following expression?

Question

How to calculate the limit without using L'hopital's rule: $$\mathrm{lim}_{x \rightarrow 0} \frac{1-\mathrm{cos} x \,\mathrm{cos}2x \,\cdots \mathrm{cos}nx}{x^{2}}\;\;?$$

Answer 1

Here's a hint for how to do it without Maclaurin expansion: $$ \begin{split} \frac{1-\cos x \cos 2x \cdots \cos nx}{x^{2}} & = \frac{1-\cos nx \, \bigl( 1 - 1 + \cos x \cdots \cos (n-1)x \bigr)}{x^{2}} \\ & = n^2 \cdot \underbrace{\frac{1-\cos nx}{(nx)^2}}_{\to 1/2} + \underbrace{\strut \cos nx}_{\to 1} \cdot \underbrace{\frac{1-\cos x \cdots \cos (n-1)x}{x^{2}}}_{\to \, \cdots \, ?} \end{split} $$ Can you continue from there?

Answer 2

\begin{gather} The\ Maclaurin\ expansion\ of\ \cos nx\ is\\ \cos nx=1-\frac{n^{2} x^{2}}{2!} +\frac{n^{4} x^{4}}{4!} +... \notag\\ Now, \notag\\ \cos x\cdotp \cos 2x=\left( 1-\frac{x^{2}}{2!} +...\right)\left( 1-\frac{2^{2} x^{2}}{2!} +..\right) \notag\\ =1-\frac{\left( 1^{2} +2^{2}\right)}{2!} x^{2} +.... \notag\\ Similarly,\ \cos x\cdotp \cos 2x\cdotp \cos 3x=\left( 1-\frac{\left( 1^{2} +2^{2}\right)}{2!} x^{2} +....\right)\left( 1-\frac{( 3x)^{2}}{2!} +..\right) \notag\\ =1-\frac{\left( 1^{2} +2^{2} +3^{2}\right) x^{2}}{2!} +... \notag\\ where\ ...\ denotes\ terms\ that\ are\ higher\ powers\ of\ x\ ( than\ 2) , \notag\\ denoted\ below\ by\ O\left( x^{4}\right) \notag\\ In\ general, \notag\\ \cos x\cos 2x\cdotp \cdotp \cdotp \cos nx=1-\frac{\left( 1^{2} +2^{2} +3^{2} +...n^{2}\right)}{2!} x^{2} +O\left( x^{4}\right) \notag\\ 1-\cos x\cos 2x\cdotp \cdotp \cdotp \cos nx=\frac{\left( 1^{2} +2^{2} +3^{2} +...n^{2}\right)}{2!} x^{2} -O\left( x^{4}\right) \notag\\ \frac{1-\cos x\cos 2x\cdotp \cdotp \cdotp \cos nx}{x^{2}} \notag\\ =\frac{\frac{\left( 1^{2} +2^{2} +3^{2} +...n^{2}\right)}{2!} x^{2} -O\left( x^{4}\right)}{x^{2}} \notag\\ =\frac{\left( 1^{2} +2^{2} +3^{2} +...n^{2}\right)}{2!} -O\left( x^{2}\right) \notag\\ Now, \notag\\ \lim _{x\rightarrow 0}\frac{1-\cos x\cos 2x\cdotp \cdotp \cdotp \cos nx}{x^{2}} =\lim _{x\rightarrow 0}\frac{\left( 1^{2} +2^{2} +3^{2} +...n^{2}\right)}{2!} -O\left( x^{2}\right) \notag\\ =\frac{\left( 1^{2} +2^{2} +3^{2} +...n^{2}\right)}{2!} \notag\\ \notag\\ \notag\\ \notag \end{gather} Can you complete the answer?