#### Find the limit of $\mathrm{ \lim _{x \rightarrow 0} \frac{1-(\cos x)(\cos 2 x)^{1 / 2}(\cos 3 x)^{1 / 3}}{x^2} }$  Option: 1 3Option: 2 5Option: 3 2Option: 4 1

The key here is to use Taylor / Maclaurin expansions as $\mathrm{x \rightarrow 0}$

$\mathrm{ \cos x=1-\frac{x^2}{2}+o\left(x^2\right) }$
and

$\mathrm{ (1+x)^n=1+n x+o(x) }$

where $\mathrm{o(f(x))}$ represents a function $\mathrm{g(x)}$ such that $\mathrm{g(x) / f(x) \rightarrow 0.}$

The starting step (based on expansion (1) above) thus needs to be written like

$\mathrm{ \lim _{x \rightarrow 0} \frac{1-\left(1-\frac{x^2}{2}+o\left(x^2\right)\right)\left(1-2 x^2+o\left(x^2\right)\right)^{1 / 2}\left(1-\frac{9 x^2}{2}+o\left(x^2\right)\right)^{1 / 3}}{x^2} }$

and the next step (using (2)) becomes

$\mathrm{ \lim _{x \rightarrow 0} \frac{1-\left(1-\frac{x^2}{2}+o\left(x^2\right)\right)\left(1-x^2+o\left(x^2\right)\right)\left(1-\frac{3 x^2}{2}+o\left(x^2\right)\right)}{x^2} }$

Finally we have via multiplication

$\mathrm{ \lim _{x \rightarrow 0} \frac{1-\left(1-3 x^2+o\left(x^2\right)\right)}{x^2}=3 }$