<s>[INST] Evaluating $\lim_{x\to 0} \frac{1-\cos x\sqrt{\cos 2x}\sqrt[3]{\cos 3x}}{x^2}$ I want to evaluate:

$$\lim_{x\to 0} \frac{1-\cos x\sqrt{\cos 2x}\sqrt[3]{\cos 3x}}{x^2}.$$

Here's what I did. We know that as soon as ${x\to 0}$ 
$$1 - \cos x = \frac{x^2}{2} + O(x^4)$$
Therefore
$$\cos x = 1 - \frac{x^2}{2} + O(x^4)$$
Now we apply this to cosines in limit
$$\cos 2x = 1 - 2x^2 + O(x^4)$$
$$\cos 3x = 1 - \frac{9x^2}{2} + O(x^4)$$
Then use equality
$$ (1 + x)^n = 1 + xn + o(x)$$
It yields us
$$\lim_{x\to 0} \frac{1 - (1 - \frac{x^2}{2})(1 - x^2)(1 - \frac{3x^2}{2})}{x^2}$$
After simplification 1s cancel and we get $-(-1/2 - 1 - 3/2) = 3$.
 [/INST] When we do the Tayor series expansion of the numerator, we don't care about the $x^3$ and higher powered terms.  They are all going to 0.  I am going to drop them as I go.
$\lim_\limits{x\to 0}\frac {1 - (1 - \frac {x^2}{2})(1-2x^2)^\frac12(1-\frac {9x^2}{2})^\frac 13}{x^2}$
Now do the binomial expansion and continue to drop the $x^3$ and higher powered terms.
$\lim_\limits{x\to 0}\frac {1 - (1 - \frac {x^2}{2})(1-x^2)(1-\frac {3x^2}{2})}{x^2}$
Multiply it out, (still dropping high powered terms)
$\lim_\limits{x\to 0}\frac {1 - (1 - (\frac 12+1+\frac 32) x^2)}{x^2}$
And evaluate.
$ 3$
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