# Remainder in taylor's theorem for multivariate functions

Excerpt is from Courant Intro Analysis book Vol 2 pg 69. I need explanation of the underlined part (entire sentence) including justification. I understand what vanishes to higher order means.

• what is the name of that book? Feb 2 '19 at 3:04
• @JimmySabater Hi i edited to include book. Feb 2 '19 at 3:15

That is the meaning of the Landau symbol, "little oh". That is, $$f(x)= o(g(x))$$ means $$\lim_{x\to0}\frac{f(x)}{g(x)}=0$$. I.e. $$f$$ goes to zero "at higher order" than $$g$$.

For instance, $$x^n=o(x^m)$$ for $$n\gt m$$.

Thus the result you have underlined follows from the fact that the remainder $$R_n$$ has higher powers of $$h$$ and $$k$$ than $$d^nf$$ does. This information is on the page above.

• How do you prove that the ratio of the two polynomials in $h$ and $k$ go to $0$? Is there are Hopital rule for multivariable functions? Also I still don't see the relevance of the very last part that $R_n=o(\sqrt{(h^2+k^2)^n})$ Feb 2 '19 at 4:05
• For the last part, try squaring both sides. One is a polynomial of degree $2n$, the other $2n+2$. Polynomials of higher degree grow (and decay) faster, a basic fact.
– user403337
Feb 2 '19 at 4:16
• What's a rough proof sketch of "Polynomials of higher degree grow (and decay) faster"? Feb 2 '19 at 4:29
• The highest degree term dominates.
– user403337
Feb 2 '19 at 5:53
• Factor it out; then compare terms of highest degree.
– user403337
Feb 2 '19 at 8:20