# L’Hôpital’s Rule—Taylor’s Theorem

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

## Abstract

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
$$\frac{{f\left( b \right) - f\left( a \right)}}{{g\left( b \right) - g\left( a \right)}} = \frac{{f\prime \left( {{x_0}} \right)}}{{g\prime \left( {{x_0}} \right)}};$$
(1.1)
(3) if f(a) ≠ f(b), then at the x0in (1.1), f′(x0) and g′(x0) are both finite.

## Keywords

Positive Integer Local Maximum Taylor Series Interior Point Nonnegative Integer
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.