Maclaurin series and Landau's big "Oh" problem.

I have found the first 5 Taylor polynomials about x=0 for cosh(x). Being p0=1, p1=1, p2=1+$\displaystyle \frac{1}{2}x^2$, p3=1+$\displaystyle \frac{1}{2}x^2$, p4=1+$\displaystyle \frac{1}{2}x^2$+$\displaystyle \frac{1}{24}x^4$.

How do I Find the Maclaurin series for cosh(x)?

I can see it should be something like

$\displaystyle f(x) \approx \sum_{n = 1}\frac{x^{2n}}{2n!}$

but how can I show a methodic way to get there from the polynomial approximations?

Also, how do I use Landau's big O notation to find

$\displaystyle \lim_{x \to 0}\frac{2cosh^2(x)-2}{1-cosh(3x)}$

I simplify this with identity $\displaystyle cosh^2(x)=\frac{1}{2}(1+cosh(2x))$to

$\displaystyle \lim_{x \to 0}\frac{cosh(2x)-1}{1-cosh(3x)}$

then change them to Maclaurin approximations with big O error.

$\displaystyle \lim_{x \to 0}\frac{2x^2+O(x^3)}{\frac{-9}{2}x^2-O(x^3)}$

How do I simplify that expression to get the limit?

Thank you for any help.