DIFFERENTIATION
Section 6.3 L’Hospital’s Rules
In this section we will discuss limit theorems that involve cases that cannot be determined by previous limit theorems. For example, iff(x) andg(x) both approach 0 asxapproaches a, then the quotientf xð Þ=g xð Þmay or may not have a limit ataand it is said to have the indeterminate form 0/0. The limit theorem for this case is due to Johann Bernoulli and first appeared in the 1696 book published by L’Hospital.
Johann Bernoulli
Johann Bernoulli (1667–1748) was born in Basel, Switzerland. Johann worked for a year in his father’s spice business, but he was not a success.
He enrolled in Basel University to study medicine, but his brother Jacob, twelve years older and a Professor of Mathematics, led him into mathematics.
Together, they studied the papers of Leibniz on the new subject of calculus.
Johann received his doctorate at Basel University and joined the faculty at Groningen in Holland, but upon Jacob’s death in 1705, he returned to Basel and was awarded Jacob’s chair in mathematics. Because of his many advances in the subject, Johann is regarded as one of the founders of calculus.
While in Paris in 1692, Johann met the Marquis Guillame Francois de L’Hospital and agreed to a financial arrangement under which he would teach the new calculus to L’Hospital, giving L’Hospital the right to use Bernoulli’s lessons as he pleased. This was subsequently continued through a series of letters. In 1696, the first book on differential calculus,L’Analyse des Infiniment Petits, was published by L’Hospital. Though L’Hospital’s name was not on the title page, his portrait was on the frontispiece and the preface states ‘‘I am indebted to the clarifications of the brothers Bernoulli, especially the younger.’’ The book contains a theorem on limits later known as L’Hospital’s Rule although it was in fact discovered by Johann Bernoulli. In 1922, manuscripts were discovered that confirmed the book consisted mainly of Bernoulli’s lessons. And in 1955, the L’Hospital–Bernoulli correspondence was published in Germany.
The initial theorem was refined and extended, and the various results are collectively referred to as L’Hospital’s (or L’H^opital’s) Rules. In this section we establish the most basic of these results and indicate how others can be derived.
Indeterminate Forms
In the preceding chapters we have often been concerned with methods of evaluating limits. It was shown in Theorem 4.2.4(b) that ifA:¼lim
x!cf xð ÞandB:¼lim
x!cg xð Þ, and ifB6¼0, then limx!c
f xð Þ g xð Þ¼ A
B :
However, ifB¼0, then no conclusion was deduced. It will be seen in Exercise 2 that if B¼0 andA6¼0, then the limit is infinite (when it exists).
The caseA¼0,B¼0 has not been covered previously. In this case, the limit of the quotientf=gis said to be ‘‘indeterminate.’’ We will see that in this case the limit may not exist or may be any real value, depending on the particular functions f and g. The symbolism 0=0 is used to refer to this situation. For example, ifais any real number, and if we definef xð Þ:¼axandg xð Þ:¼x, then
x!lim0
f xð Þ g xð Þ¼lim
x!0
ax x ¼lim
x!0a¼a:
Thus the indeterminate form 0=0 can lead to any real numberaas a limit.
Other indeterminate forms are represented by the symbols 1=1; 0 1;
00;11;10; and1 1. These notations correspond to the indicated limiting behavior and juxtaposition of the functionsfandg. Our attention will be focused on the indeterminate forms 0=0 and1=1. The other indeterminate cases are usually reduced to the form 0=0 or 1=1by taking logarithms, exponentials, or algebraic manipulations.
A Preliminary Result
To show that the use of differentiation in this context is a natural and not surprising development, we first establish an elementary result that is based simply on the definition of the derivative.
6.3.1 Theorem Let f and g be defined on[a, b], let f að Þ ¼g að Þ ¼0, and let g xð Þ 6¼0for a<x<b.If f and g are differentiable at a and if g0ð Þ 6¼a 0, then the limit of f=g at a exists and is equal to f0ð Þ=ga 0ð Þ. Thusa
x!aþlim f xð Þ g xð Þ¼f0ð Þa
g0ð Þa :
Proof. Since f að Þ ¼g að Þ ¼0, we can write the quotient f xð Þ=g xð Þ for a<x<b as follows:
f xð Þ
g xð Þ¼f xð Þ f að Þ g xð Þ g að Þ¼
f xð Þ f að Þ xa g xð Þ g að Þ
xa : Applying Theorem 4.2.4(b), we obtain
x!aþlim f xð Þ g xð Þ¼ lim
x!aþ
f xð Þ f að Þ xa
x!aþlim
g xð Þ g að Þ xa
¼f0ð Þa g0ð Þa :
Q.E.D.
6.3 L’HOSPITAL’S RULES 181
Warning The hypothesis that f að Þ ¼g að Þ ¼0 is essential here. For example, if f xð Þ :¼xþ17 andg xð Þ:¼2xþ3 forx2R, then
x!lim0
f xð Þ g xð Þ¼17
3 ; while f0ð Þ0 g0ð Þ0 ¼1
2: The preceding result enables us to deal with limits such as
limx!0
x2þx
sin 2x ¼20þ1 2 cos 0 ¼1
2:
To handle limits wherefandgare not differentiable at the pointa, we need a more general version of the Mean Value Theorem due to Cauchy.
6.3.2 Cauchy Mean Value Theorem Let f and g be continuous on [a, b] and differentiable on(a, b), and assume that g0ð Þ 6¼x 0 for all x in(a, b). Then there exists c in(a, b)such that
f bð Þ f að Þ g bð Þ g að Þ¼f0ð Þc
g0ð Þc :
Proof. As in the proof of the Mean Value Theorem, we introduce a function to which Rolle’s Theorem will apply. First we note that sinceg0ð Þ 6¼x 0 for allxin (a,b), it follows from Rolle’s Theorem that g að Þ 6¼g bð Þ. Forxin [a,b], we now define
h xð Þ:¼f bð Þ f að Þ
g bð Þ g að Þðg xð Þ g að ÞÞ ðf xð Þ f að ÞÞ:
Thenhis continuous on [a, b], differentiable on (a, b), andh að Þ ¼h bð Þ ¼0. Therefore, it follows from Rolle’s Theorem 6.2.3 that there exists a pointcin (a, b) such that
0¼h0ð Þ ¼c f bð Þ f að Þ
g bð Þ g að Þg0ð Þ c f0ð Þ:c
Since g0ð Þ 6¼c 0, we obtain the desired result by dividing byg0ð Þc. Q.E.D.
Remarks The preceding theorem has a geometric interpretation that is similar to that of the Mean Value Theorem 6.2.4. The functionsfandgcan be viewed as determining a curve in the plane by means of the parametric equationsx¼f tð Þ; y¼g tð Þwhereatb. Then the conclusion of the theorem is that there exists a pointðf cð Þ;g cð ÞÞon the curve for somecin (a, b) such that the slopeg0ð Þ=fc 0ð Þc of the line tangent to the curve at that point is equal to the slope of the line segment joining the endpoints of the curve.
Note that if g xð Þ ¼x, then the Cauchy Mean Value Theorem reduces to the Mean Value Theorem 6.2.4.
L’Hospital’s Rule, I
We will now establish the first of L’Hospital’s Rules. For convenience, we will consider right-hand limits at a pointa; left-hand limits, and two-sided limits are treated in exactly the same way. In fact, the theorem even allows the possibility thata¼ 1. The reader should observe that, in contrast with Theorem 6.3.1, the following result does not assume the differentiability of the functions at the point a. The result asserts that the limiting behavior off xð Þ=g xð Þasx!aþis the same as the limiting behavior off0ð Þ=gx 0ð Þx as
x!aþ, including the case where this limit is infinite. An important hypothesis here is that bothfandgapproach 0 asx!aþ.
6.3.3 L’Hospital’s Rule, I Let1 a<b 1and let f, g be differentiable on(a,b) such that g0ð Þ 6¼x 0 for all x2ða;bÞ. Suppose that
ð1Þ lim
x!aþf xð Þ ¼0¼ lim
x!aþg xð Þ:
(a) If lim
x!aþ
f0ð Þx
g0ð Þx ¼L2R;then lim
x!aþ
f xð Þ g xð Þ¼L:
(b) If lim
x!aþ
f0ð Þx
g0ð Þx ¼L2 1;f 1g;then lim
x!aþ
f xð Þ g xð Þ¼L:
Proof. Ifa<a<b<b, then Rolle’s Theorem implies thatgð Þ 6¼b gð Þa. Further, by the Cauchy Mean Value Theorem 6.3.2, there existsu2ða;bÞsuch that
ð2Þ fð Þ b fð Þa
gð Þ b gð Þa ¼f0ð Þu g0ð Þu :
Case (a): IfL2R and ife>0 is given, there existsc2ða;bÞsuch that Le < f0ð Þu
g0ð Þu <Lþe for u2ða;cÞ;
whence it follows from (2) that ð3Þ Le < fð Þ b fð Þa
gð Þ b gð Þa <Lþe for a<a<bc:
If we take the limit in (3) asa!aþ, we have Le fð Þb
gð Þb Lþe for b2 ða;c:
Sincee>0 is arbitrary, the assertion follows.
Case (b): IfL¼ þ1and ifM>0 is given, there existsc2ða;bÞsuch that f0ð Þu
g0ð Þu >M for u2ða;cÞ;
whence it follows from (2) that ð4Þ fð Þ b fð Þa
gð Þ b gð Þa >M for a<a<b<c:
If we take the limit in (4) asa!aþ, we have fð Þb
gð Þb M for b2ða;cÞ:
SinceM >0 is arbitrary, the assertion follows.
IfL¼ 1, the argument is similar. Q.E.D.
The corresponding theorem for left-hand limits is readily proved in the same manner.
The result for two-sided limits then follows immediately if both one-sided limits exist and 6.3 L’HOSPITAL’S RULES 183