James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

314 Chapter 4 Applications of Differentiation
86. Suppose f is a positive function. If lim xl a f sxd − 0 and
lim xl a tsxd − `, show that
lim f f sxdg tsxd − 0
x l a
This shows that 0` is not an indeterminate form.
87. If f 9 is continuous, f s2d − 0, and f 9s2d − 7, evaluate
lim
x l 0
f s2 1 3xd 1 f s2 1 5xd
x
88. For what values of a and b is the following equation true?
sin 2x
lim
x l 0S 2D 1 a 1 b − 0
x 3 x
89. If f 9 is continuous, use l’Hospital’s Rule to show that
lim
h l 0
f sx 1 hd 2 f sx 2 hd
2h
− f 9sxd
Explain the meaning of this equation with the aid of a
diagram.
90. If f 0 is continuous, show that
91. Let
; 92. Let
lim
h l 0
f sx 1 hd 2 2f sxd 1 f sx 2 hd
h 2
f sxd −H 2
e21yx 0
if x ± 0
if x − 0
− f 0sxd
(a) Use the definition of derivative to compute f 9s0d.
(b) Show that f has derivatives of all orders that are defined
on R. [Hint: First show by induction that there is a
poly nomial p nsxd and a nonnegative integer k n such that
f snd sxd − p nsxdf sxdyx kn for x ± 0.]
f sxd −H | x | x
1
if x ± 0
if x − 0
(a) Show that f is continuous at 0.
(b) Investigate graphically whether f is differentiable at 0
by zooming in several times toward the point s0, 1d on
the graph of f .
(c) Show that f is not differentiable at 0. How can you
reconcile this fact with the appearance of the graphs in
part (b)?
writing Project
Thomas Fisher Rare Book Library
www.stewartcalculus.com
The Internet is another source of
information for this project. Click on
History of Mathematics for a list of
reliable websites.
The origins of l’hospital’s rule
L’Hospital’s Rule was first published in 1696 in the Marquis de l’Hospital’s calculus textbook
Analyse des Infiniment Petits, but the rule was discovered in 1694 by the Swiss mathematician
John (Johann) Bernoulli. The explanation is that these two mathematicians had entered into a
curious business arrangement whereby the Marquis de l’Hospital bought the rights to Bernoulli’s
mathematical discoveries. The details, including a translation of l’Hospital’s letter to
Bernoulli pro posing the arrangement, can be found in the book by Eves [1].
Write a report on the historical and mathematical origins of l’Hospital’s Rule. Start by providing
brief biographical details of both men (the dictionary edited by Gillispie [2] is a good
source) and outline the business deal between them. Then give l’Hospital’s state ment of his
rule, which is found in Struik’s sourcebook [4] and more briefly in the book of Katz [3]. Notice
that l’Hospital and Bernoulli formulated the rule geometrically and gave the answer in terms of
differentials. Compare their statement with the version of l’Hospital’s Rule given in Section 4.4
and show that the two statements are essentially the same.
1. Howard Eves, In Mathematical Circles (Volume 2: Quadrants III and IV) (Boston: Prindle,
Weber and Schmidt, 1969), pp. 20–22.
2. C. C. Gillispie, ed., Dictionary of Scientific Biography (New York: Scribner’s, 1974). See
the article on Johann Bernoulli by E. A. Fellmann and J. O. Fleckenstein in Volume II and
the article on the Marquis de l’Hospital by Abraham Robinson in Volume VIII.
3. Victor Katz, A History of Mathematics: An Introduction (New York: HarperCollins, 1993),
p. 484.
4. D. J. Struik, ed., A Sourcebook in Mathematics, 1200–1800 (Princeton, NJ: Princeton
University Press, 1969), pp. 315–16.
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