Let $f : \mathbb R^3 → \mathbb R$ be the function defined by $f(x, y, z) := e^{x^2+y+xz}cos(x)$.
Compute the first order Taylor polynomial about $(0, 0, 0)$, the second order Taylor polynomial about $(0, 0, 0)$, and the remainder of order $3$ about $(0, 0, 0)$.
Deduce from the previous results that lim$_{(x,y,z)→(0,0,0)}$ $\frac{e^{x^2+y+xz}cos(x)−(1+y+ 1/2(x^2+y^2+2xz))} {\sqrt{x^2+y^2+z^2}} = 0$· Justify your answer.
My attempts:
I found $P_1(x) = 1+y$
$P_2(x) = 1+y+1/2(x^2+y^2+2xz)$
But I don't know what is the formula for computing the remainder of order 3 about $(0,0,0)$. Can someone please write it down?
Also concerning the limit, do I say that $e^{x^2+y+xz}cos(x) = P_2(x)$ and proceed from there? If not, what shall I do? Thank you for the help.