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(11.5.1) L'HÔPITAL'S RULE (0/0) Suppose that f(x) → 0 and g(x) → 0 and in the approach g(x) and g(x) are never 0. f'(x) f(x) If →Y, then → Y. g'(x) g(x) THEOREM 11.5.2 THE CAUCHY MEAN-VALUE THEOREM* Suppose that f and g are differentiable on (a, b) and continuous on [a, b]. If g is never 0 in (a, b), then there is a number rin (a, b) for which f'(r) f(b) - f(a) g'(r) = g(b)g(a)

(11.5.1) L'HÔPITAL'S RULE (0/0) Suppose that f(x) → 0 and g(x) → 0 and in the approach g(x) and g(x) are never 0. f'(x) f(x) If →Y, then → Y. g'(x) g(x) THEOREM 11.5.2 THE CAUCHY MEAN-VALUE THEOREM* Suppose that f and g are differentiable on (a, b) and continuous on [a, b]. If g is never 0 in (a, b), then there is a number rin (a, b) for which f'(r) f(b) - f(a) g'(r) = g(b)g(a)

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Practice using 11.5.2 to derive 11.5.1

(11.5.1) L'HÔPITAL'S RULE (0/0) Suppose that f(x) → 0 and g(x) → 0 and in the approach g(x) and g(x) are never 0. f'(x) f(x) If →Y, then → Y. g'(x) g(x)
Transcribed Image Text:(11.5.1) L'HÔPITAL'S RULE (0/0) Suppose that f(x) → 0 and g(x) → 0 and in the approach g(x) and g(x) are never 0. f'(x) f(x) If →Y, then → Y. g'(x) g(x)
THEOREM 11.5.2 THE CAUCHY MEAN-VALUE THEOREM* Suppose that f and g are differentiable on (a, b) and continuous on [a, b]. If g is never 0 in (a, b), then there is a number rin (a, b) for which f'(r) f(b) - f(a) g'(r) = g(b)g(a)
Transcribed Image Text:THEOREM 11.5.2 THE CAUCHY MEAN-VALUE THEOREM* Suppose that f and g are differentiable on (a, b) and continuous on [a, b]. If g is never 0 in (a, b), then there is a number rin (a, b) for which f'(r) f(b) - f(a) g'(r) = g(b)g(a)
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