L’Hôpital’s Rule—Taylor’s Theorem

  • Emanuel Fischer
Part of the Undergraduate Texts in Mathematics book series (UTM)


Theorem 1.1 (Cauchy’s Mean-Value Theorem). If f and g are real-valued functions of a real variable, both continuous on the bounded closed interval [a,b], differentiable in the extended sense on (a; b) with g′(x) ≠ 0 for x ∈ (a; b), having derivatives which are not simultaneously infinite, then (1) g(a) ≠ g(b); (2) there exists an x0 ∈ (a; b) such that
$$\frac{{f\left( b \right) - f\left( a \right)}}{{g\left( b \right) - g\left( a \right)}} = \frac{{f\prime \left( {{x_0}} \right)}}{{g\prime \left( {{x_0}} \right)}};$$
(3) if f(a) ≠ f(b), then at the x0in (1.1), f′(x0) and g′(x0) are both finite.


Positive Integer Local Maximum Taylor Series Interior Point Nonnegative Integer 
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Copyright information

© Springer-Verlag New York, Inc. 1983

Authors and Affiliations

  • Emanuel Fischer
    • 1
  1. 1.Department of MathematicsAdelphi UniversityGarden CityUSA

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