MATH115 - Indeterminate Forms and Improper Integrals

MATH115Indeterminate Forms and Improper Integrals

Paolo Lorenzo Bautista

De La Salle University

June 24, 2014

PLBautista (DLSU) MATH115 June 24, 2014 1 / 25

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.



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.



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)

.



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.



Guillaume de l’Hopital (1661-1704)



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.



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



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.



RemarkOther indeterminate forms are the following:

1.±∞±∞

2. 0 · (∞)3. ∞+∞4. 00

5. (±∞)0

6. 1±∞



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.




1. limx→+∞

sin 2x

1x

2. limx→+∞

1− e1/x

−3x

3. limx→+∞

1x

tan 2x



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




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.


x→+∞f (x) is +∞ or −∞ and lim

x→+∞g(x) is +∞ or −∞.

If limx→+∞

f ′(x)g′(x)


f (x)g(x)

= L.




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.


x→+∞f (x) is +∞ or −∞ and lim

x→+∞g(x) is +∞ or −∞.

If limx→+∞

f ′(x)g′(x)


f (x)g(x)

= L.




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



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]


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.


Improper Integrals


DefinitionIf f is continuous for all x ≥ a, then∫ b

−∞f (x)dx = lim

a→−∞

∫ b

af (x)dx



Improper Integrals


RemarkIf the aforementioned limits exist, then the improper integral is said tobe convergent. Otherwise, the improper integral is divergent.


Improper Integrals

ExampleEvaluate the following improper integrals:

1.∫ 2

−∞

dx(4− x)2

2.∫ +∞

0xe−xdx

3.∫ +∞

0sin xdx


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


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



Improper Integrals


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



Improper Integrals


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.


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


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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MATH115 - Indeterminate Forms and Improper Integrals