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MATH
[1251]

Propositions from Taylor's theorem

All propositions from Taylor's theorem.

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Comparing the first nonzero Taylor terms

Suppose m≥1m\ge1, ff and gg are mm times differentiable near aa, f(j)(a)=g(j)(a)=0f^{(j)}(a)=g^{(j)}(a)=0 for 0≤j<m0\le j<m and f(m)(a),g(m)(a)≠0f^{(m)}(a),g^{(m)}(a)\ne0. Then

lim⁡x→af(x)g(x)=f(m)(a)g(m)(a).\lim_{x\to a}\frac{f(x)}{g(x)} =\frac{f^{(m)}(a)}{g^{(m)}(a)}.

The leading expansions are

f(x)=f(m)(a)m!(x−a)m+o((x−a)m),f(x)=\frac{f^{(m)}(a)}{m!}(x-a)^m+o((x-a)^m),

and

g(x)=g(m)(a)m!(x−a)m+o((x−a)m).g(x)=\frac{g^{(m)}(a)}{m!}(x-a)^m+o((x-a)^m).

The common power (x−a)m(x-a)^m cancels in the quotient. This explains why repeated L'Hôpital differentiation can work, but the hypotheses of that rule must still be checked.

  1. [1]:Taylor's theorem with remainder

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Comparing the first nonzero Taylor terms