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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.

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  • $\begingroup$ what is the name of that book? $\endgroup$
    – James
    Feb 2 '19 at 3:04
  • $\begingroup$ @JimmySabater Hi i edited to include book. $\endgroup$
    – helios321
    Feb 2 '19 at 3:15
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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.

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  • $\begingroup$ 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 $R_n=o(\sqrt{(h^2+k^2)^n})$ $\endgroup$
    – helios321
    Feb 2 '19 at 4:05
  • $\begingroup$ 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. $\endgroup$
    – user403337
    Feb 2 '19 at 4:16
  • $\begingroup$ What's a rough proof sketch of "Polynomials of higher degree grow (and decay) faster"? $\endgroup$
    – helios321
    Feb 2 '19 at 4:29
  • $\begingroup$ The highest degree term dominates. $\endgroup$
    – user403337
    Feb 2 '19 at 5:53
  • $\begingroup$ Factor it out; then compare terms of highest degree. $\endgroup$
    – user403337
    Feb 2 '19 at 8:20

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