L’Hôpital’s Rule—Taylor’s Theorem

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 x₀ in (1.1), f′(x₀) and g′(x₀) are both finite.