I tried to taylor expand a function of three variables $f(x,y,z)$ around $y=0,z=0$ but I don't find a way to decide where to stop the expansion.
Usually when one expands a function of one variable $g(x)$ around $x=0$
$ g(0)+\partial_x g(0) \cdot x + \frac 1 2 \partial^2_x g(0) \cdot x^2 + \mathcal O (x^3) $
one can immediately say that contributions of (say) $x^3$ or higher can be neglected if $x^3$ is already smaller than the desired precision.
But in the case of three variables I get an expansion like
$ f(x,0,0) + \partial_yf(x,0,0)\cdot y + \partial_zf(x,0,0)\cdot z+\frac 1 2 \big(\partial_y^2f(x,0,0)\cdot y^2 + 2\partial_y\partial_z f(x,0,0)\cdot yz + \partial^2_z f(x,0,0)\cdot z^2\big) + ... $
but I don't have any clue where to stop. I think the reason is that there are still functions on $x$ to which I can't say anything about their order in opposite to the one dimensional case where just constants appear.
So for example when I say that both $y$ and $z$ are very small but $z$ is way smaller than $y$ and I keep a term like $\partial^2_z f(x,0,0)\cdot z^2$ then there is no reason to neglect the term (say) $\partial_y^7\partial_zf(x,0,0)\cdot y^7z$ since the latter can be bigger than the former for some $x$.
Can someone help me out?