limx→0 1x121−cos⁡x22−cos⁡x44+cos⁡x22cos⁡x44 is equal to

A
$\frac{1}{32}$
B
$\frac{1}{256}$
C
$\frac{1}{16}$
D
$- \frac{1}{256}$

Solution:

$lim_{x \to 0} \frac{1}{x^{12}} \{1 - \cos \frac{x^{2}}{2} - \cos \frac{x^{4}}{4} + \cos \frac{x^{2}}{2} \cos \frac{x^{4}}{4}\}$

$= lim_{x \to 0} \frac{(1 - \cos \frac{x^{2}}{2}) (1 - \cos \frac{x^{4}}{4})}{x^{12}}$

$\begin{array}{l} = lim_{x \to 0} \frac{1 - \cos \frac{x^{2}}{2}}{{(\frac{x^{2}}{2})}^{2} \times 2^{2}} \times \frac{1 - \cos (\frac{x^{4}}{4})}{{(\frac{x^{4}}{4})}^{2} \times 4^{2}} \\ = \frac{1}{2} \times \frac{1}{2^{2}} \times \frac{1}{2} \times \frac{1}{4^{2}} = \frac{1}{256} [∵ lim_{α \to 0} \frac{1 - \cos α}{α^{2}} = \frac{1}{2}] \end{array}$

$lim_{x \to 0} \frac{1}{x^{12}} \{1 - \cos \frac{x^{2}}{2} - \cos \frac{x^{4}}{4} + \cos \frac{x^{2}}{2} \cos \frac{x^{4}}{4}\}$ is equal to

Solution:

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