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.

PLBautista (DLSU) MATH115 June 24, 2014 2 / 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.

PLBautista (DLSU) MATH115 June 24, 2014 2 / 25

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)

.

PLBautista (DLSU) MATH115 June 24, 2014 3 / 25

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.

PLBautista (DLSU) MATH115 June 24, 2014 4 / 25

Indeterminate Forms and L’Hopital’s Rule Cauchy Mean-Value Theorem

Guillaume de l’Hopital (1661-1704)

PLBautista (DLSU) MATH115 June 24, 2014 5 / 25

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.

PLBautista (DLSU) MATH115 June 24, 2014 6 / 25

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

PLBautista (DLSU) MATH115 June 24, 2014 7 / 25

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.

PLBautista (DLSU) MATH115 June 24, 2014 8 / 25

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±∞

PLBautista (DLSU) MATH115 June 24, 2014 9 / 25

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.

PLBautista (DLSU) MATH115 June 24, 2014 10 / 25

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

PLBautista (DLSU) MATH115 June 24, 2014 11 / 25

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

PLBautista (DLSU) MATH115 June 24, 2014 12 / 25

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.

PLBautista (DLSU) MATH115 June 24, 2014 13 / 25

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.

PLBautista (DLSU) MATH115 June 24, 2014 13 / 25

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

PLBautista (DLSU) MATH115 June 24, 2014 14 / 25

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]

PLBautista (DLSU) MATH115 June 24, 2014 15 / 25

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.

PLBautista (DLSU) MATH115 June 24, 2014 16 / 25

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.

PLBautista (DLSU) MATH115 June 24, 2014 17 / 25

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.

PLBautista (DLSU) MATH115 June 24, 2014 18 / 25

Improper Integrals

ExampleEvaluate the following improper integrals:

1.∫ 2

−∞

dx(4− x)2

2.∫ +∞

0xe−xdx

3.∫ +∞

0sin xdx

PLBautista (DLSU) MATH115 June 24, 2014 19 / 25

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

PLBautista (DLSU) MATH115 June 24, 2014 20 / 25

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.

PLBautista (DLSU) MATH115 June 24, 2014 21 / 25

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.

PLBautista (DLSU) MATH115 June 24, 2014 22 / 25

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.

PLBautista (DLSU) MATH115 June 24, 2014 23 / 25

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

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

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

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