- #1

Adgorn

- 130

- 18

- Homework Statement
- (Spivak's Calculus, 20-9) This is a part of a problem to express the Taylor polynomial of a composite function. Let ##f(x)=P_{n,0,f}(x)+R_{n,0,f}(x)## and ##g(x)=P_{n,0,g}(x)+R_{n,0,g}(x)## where ##P_{n,0,f},P_{n,0,g}## are the Taylor polynomials of degree ##n## at ##0## for ##f## and ##g##, ##R_{n,0,f},R_{n,0,g}## are the corresponding remainders, and ##g(0)=0##. In part of the problem I need to show that $$\lim_{x \rightarrow 0} \frac {R_{n,0,f}(g(x))} {x^n}=0$$

- Relevant Equations
- If ##p## and ##q## are polynomials in ##x-a## and ##\lim_{x \rightarrow 0} \frac {R(x)} {(x-a)^n}=0## then ##p(q(x)+R(x))=p(q(x))+\bar R(x)## where ##\lim_{x \rightarrow 0} \frac {\bar R(x)} {(x-a)^n}=0##

Since $$\lim_{x \rightarrow 0} \frac {R_{n,0,f}(x)} {x^n}=0,$$ ##P_{n,0,g}(x)## contains only terms of degree ##\geq 1## and ##R_{n,0,g}## approaches ##0## as quickly as ##x^n##, I can most likely prove this using ##\epsilon - \delta## arguments, but that seems overly complicated. I also can't use Taylor's Theorem since I don't know if ##f^{(n+1)}## exists. L'hopital's rule also didn't seem to do much since I know pretty much nothing about the derivatives of ##g## or of ##R_{n,0,f}##, so I don't know if ##R_{n,0,f}(g(x))^{(n)}## approaches #0# at #0#, as a result I'm not really sure how I'm supposed to do this. Help would be appreciated.

Also this is in the chapter that introduces Taylor Polynomials, I've not reached anything about infinite series yet so please no solutions which involve infinite sums or expansions. Thanks in advance to the helpers :)

Also this is in the chapter that introduces Taylor Polynomials, I've not reached anything about infinite series yet so please no solutions which involve infinite sums or expansions. Thanks in advance to the helpers :)