I want to find values for constants $C,D$ so that $$ e^{cosh(h)-1}+\frac{1}{C}+\frac{h^2}{D}=O(h^4) $$ holds.
I know that by definition, for $cosh(h)-1\rightarrow 0$, we have
$$ e^{cosh(h)-1}=1+(cosh(h)-1)+O(cosh(h)-1)^2) $$ $$ =cosh(h)+O(cosh^2(h)-2cosh(h)+1) $$ After that, I can see that $$ \frac{1}{C}+\frac{h^2}{D}=\frac{D+Ch^2}{CD} $$ but I'm not really sure where to go after that.