<s>[INST] Computing $\lim_{x\to0} \frac 8 {x^8} \left[ 1 - \cos\frac{x^2} 2 - \cos\frac{x^2}4 + \cos\frac{x^2}2\cos\frac{x^2}4 \right]$ without using L'Hospital We have to find the following limit.
$$\lim_{x\to0} \frac 8 {x^8} \left[ 1 - \cos\frac{x^2} 2 - \cos\frac{x^2}4 + \cos\frac{x^2}2\cos\frac{x^2}4 \right]$$
In this I thought to use Lhopital . But using that it will become too long . Is there ny short method .
 [/INST] HINT:
$$1-a-b+ab=(1-a)(1-b)$$
$$\lim_{h\to0}\frac{1-\cos2h}{h^2}=2\lim_{h\to0}\left(\frac{\sin h}h\right)^2 = \text{?}$$
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