math115 - indeterminate forms and improper integrals
TRANSCRIPT
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MATH115Indeterminate Forms and Improper Integrals
Paolo Lorenzo Bautista
De La Salle University
June 24, 2014
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Indeterminate Forms and L’Hopital’s Rule Cauchy Mean-Value Theorem
Theorem (Mean-Value Theorem)Let f be a function that satisfies both of the following statements:
i. f is continuous on the closed interval [a, b].ii. f is differentiable on the open interval (a, b).
Then there is a number c ∈ (a, b) such that
f ′(c) =f (b)− f (a)
b− a.
Remark: When f (a) = f (b), we have a special case of the MVT, calledRolle’s Theorem.
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Indeterminate Forms and L’Hopital’s Rule Cauchy Mean-Value Theorem
Theorem (Mean-Value Theorem)Let f be a function that satisfies both of the following statements:
i. f is continuous on the closed interval [a, b].ii. f is differentiable on the open interval (a, b).
Then there is a number c ∈ (a, b) such that
f ′(c) =f (b)− f (a)
b− a.
Remark: When f (a) = f (b), we have a special case of the MVT, calledRolle’s Theorem.
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Indeterminate Forms and L’Hopital’s Rule Cauchy Mean-Value Theorem
Theorem (Cauchy’s Mean-Value Theorem)Let f and g be two functions that satisfies the following statements:
i. f and g are continuous on the closed interval [a, b].ii. f and g are differentiable on the open interval (a, b).
iii. For all x in the open interval (a, b), g′(x) 6= 0.Then there is a number z ∈ (a, b) such that
f (b)− f (a)g(b)− g(a)
=f ′(z)g′(z)
.
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Indeterminate Forms and L’Hopital’s Rule Cauchy Mean-Value Theorem
ExampleFind all values of z in the interval (0, 1) satisfying the conclusion ofCauchy’s Mean-Value Thoerem for the functions f (x) = 2x2 + 3x− 4and g(x) = 2x3 − 8x + 3.
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Indeterminate Forms and L’Hopital’s Rule Cauchy Mean-Value Theorem
Guillaume de l’Hopital (1661-1704)
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Indeterminate Forms and L’Hopital’s Rule Cauchy Mean-Value Theorem
L’Hopital’s Rule
Theorem (L’Hopital’s Rule)Let f and g be functions differentiable on an open interval I, exceptpossibly at the number a in I. Suppose that for all x 6= a in I, g′(x) 6= 0.Suppose further that lim
x→af (x) = 0 and lim
x→ag(x) = 0.
If limx→a
f ′(x)g′(x)
= L, then limx→a
f (x)g(x)
= L.
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Indeterminate Forms and L’Hopital’s Rule Cauchy Mean-Value Theorem
ExampleEvaluate the following limits:
1. limx→0
sin xx
2. limx→1
x2 − 1x− 1
3. limx→0
tan x− xx− sin x
4. limx→1
ln xx− 1
5. limθ→0
θ − sin θtan3 θ
6. limx→0
ex − 10x
x
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Indeterminate Forms and L’Hopital’s Rule Cauchy Mean-Value Theorem
Indeterminate Forms
DefinitionIf f and g are two functions such that
limx→a
f (x) = 0 and limx→a
g(x) = 0,
thenf (x)g(x)
has the indeterminate form00
at a.
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Indeterminate Forms and L’Hopital’s Rule Cauchy Mean-Value Theorem
RemarkOther indeterminate forms are the following:
1.±∞±∞
2. 0 · (∞)3. ∞+∞4. 00
5. (±∞)0
6. 1±∞
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Indeterminate Forms and L’Hopital’s Rule Cauchy Mean-Value Theorem
Theorem (L’Hopital’s Rule)Let f and g be functions differentiable for all x > N, where N is apositive constant. Suppose that for all x > N, g′(x) 6= 0. Supposefurther that lim
x→+∞f (x) = 0 and lim
x→+∞g(x) = 0.
If limx→+∞
f ′(x)g′(x)
= L, then limx→+∞
f (x)g(x)
= L.
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Indeterminate Forms and L’Hopital’s Rule Cauchy Mean-Value Theorem
ExampleEvaluate the following limits:
1. limx→+∞
sin 2x
1x
2. limx→+∞
1− e1/x
−3x
3. limx→+∞
1x
tan 2x
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Indeterminate Forms and L’Hopital’s Rule Cauchy Mean-Value Theorem
ExerciseEvaluate the following limits:
1. limx→2
sinπx2− x
2. limx→0
sin2 xsin x2
3. limx→0
tan 3xtan 2x
4. limx→π/2
ln(sin x)(π − 2x)2
5. limx→0
(1 + x)1/5 − (1− x)1/5
(1 + x)1/3 − (1− x)1/3
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Indeterminate Forms and L’Hopital’s Rule Cauchy Mean-Value Theorem
Theorem (L’Hopital’s Rule)Let f and g be functions differentiable on an open interval I, exceptpossibly at the number a in I. Suppose that for all x 6= a in I, g′(x) 6= 0.Suppose further that lim
x→af (x) is +∞ or −∞ and lim
x→ag(x) is +∞ or −∞.
If limx→a
f ′(x)g′(x)
= L, then limx→a
f (x)g(x)
= L.
Theorem (L’Hopital’s Rule)Let f and g be functions differentiable for all x > N, where N is apositive constant. Suppose that for all x > N, g′(x) 6= 0. Supposefurther that lim
x→+∞f (x) is +∞ or −∞ and lim
x→+∞g(x) is +∞ or −∞.
If limx→+∞
f ′(x)g′(x)
= L, then limx→+∞
f (x)g(x)
= L.
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Indeterminate Forms and L’Hopital’s Rule Cauchy Mean-Value Theorem
Theorem (L’Hopital’s Rule)Let f and g be functions differentiable on an open interval I, exceptpossibly at the number a in I. Suppose that for all x 6= a in I, g′(x) 6= 0.Suppose further that lim
x→af (x) is +∞ or −∞ and lim
x→ag(x) is +∞ or −∞.
If limx→a
f ′(x)g′(x)
= L, then limx→a
f (x)g(x)
= L.
Theorem (L’Hopital’s Rule)Let f and g be functions differentiable for all x > N, where N is apositive constant. Suppose that for all x > N, g′(x) 6= 0. Supposefurther that lim
x→+∞f (x) is +∞ or −∞ and lim
x→+∞g(x) is +∞ or −∞.
If limx→+∞
f ′(x)g′(x)
= L, then limx→+∞
f (x)g(x)
= L.
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Indeterminate Forms and L’Hopital’s Rule Cauchy Mean-Value Theorem
ExampleEvaluate the following limits:
1. limx→+∞
x2
ex
2. limx→0+
tan x(ln x)
3. limx→1
(1
ln x− 1
x− 1
)4. lim
x→0+xsin x
5. limx→+∞
(x2 −√
x4 − x2 + 2)
6. limx→0
(1 + 3x)1/x
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Indeterminate Forms and L’Hopital’s Rule Cauchy Mean-Value Theorem
ExerciseEvaluate the following limits:
1. limx→1/2−
ln(1− 2x)tanπx
2. limx→+∞
(ex + x)2/x
3. limx→0+
(sin x)x2
4. limx→0
[(cos x)ex2/2]4/x4
5. limx→+∞
[(x6 + 3x5 + 4)1/6 − x]
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Improper Integrals
Improper Integrals with Infinite Limits of Integration
DefinitionIf f is continuous for all x ≥ a, then∫ +∞
af (x)dx = lim
b→+∞
∫ b
af (x)dx
if this limit exists.
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Improper Integrals
Improper Integrals with Infinite Limits of Integration
DefinitionIf f is continuous for all x ≥ a, then∫ b
−∞f (x)dx = lim
a→−∞
∫ b
af (x)dx
if this limit exists.
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Improper Integrals
Improper Integrals with Infinite Limits of Integration
RemarkIf the aforementioned limits exist, then the improper integral is said tobe convergent. Otherwise, the improper integral is divergent.
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Improper Integrals
ExampleEvaluate the following improper integrals:
1.∫ 2
−∞
dx(4− x)2
2.∫ +∞
0xe−xdx
3.∫ +∞
0sin xdx
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Improper Integrals
ExerciseEvaluate the following improper integrals:
1.∫ +∞
0e−x/3dx
2.∫ 0
−∞x5−x2
dx
3.∫ +∞
0x2−xdx
4.∫ +∞
5
xdx3√
9− x2
5.∫ +∞
−∞e−|x|dx
6.∫ +∞
e
dxx(ln x)2
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Improper Integrals
Improper Integrals with an Infinite Discontinuity
DefinitionIf f is continuous for all x in the half open interval (a, b], and iflim
x→a+|f (x)| = +∞, then
∫ b
af (x)dx = lim
t→a+
∫ b
tf (x)dx
if this limit exists.
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Improper Integrals
Improper Integrals with an Infinite Discontinuity
DefinitionIf f is continuous for all x in the half open interval [a, b), and iflim
x→b−|f (x)| = +∞, then
∫ b
af (x)dx = lim
t→b−
∫ t
af (x)dx
if this limit exists.
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Improper Integrals
Improper Integrals with an Infinite Discontinuity
DefinitionIf f is continuous for all x in the interval [a, b] except at c wherea < c < b, and if lim
x→c|f (x)| = +∞, then
∫ b
af (x)dx = lim
t→a+
∫ b
tf (x)dx + lim
s→b−
∫ s
af (x)dx
if both these limits exist.
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Improper Integrals
ExampleEvaluate the following improper integrals:
1.∫ 1
0
dx√1− x
2.∫ 1
0x ln xdx
3.∫ −3
−5
xdx√x2 − 9
4.∫ +∞
0
dxx3
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Improper Integrals
ExerciseEvaluate the following improper integrals:
1.∫ −3
−5
dw(w + 1)1/3
2.∫ 2
−2
dxx3
3.∫ 2
1/2
dzz(ln z)1/5
4.∫ π/2
0
dy1− sin y
5.∫ +∞
2
dxx√
x2 − 4
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