Lecture 19Section 10.5 Indeterminate Form (0/0)
Section
10.6 Other Indeterminate Forms (∞/∞), (0 · ∞), · · ·
Jiwen He
1
Indeterminate Forms
1.1
Indeterminate Form (0/0)
Exampleis 1.
What
the Indeterminate Form (0/0)?
x−2
1
1
1
x−2
= lim
= lim
=
=
2
x→2
x→2
x −4
(x − 2)(x + 2)
x+2
2+2
4
f 0 (x)
1
1
1
= lim
=
=
(Amazing!!)
(L’Hôpital’s Rule) = lim 0
x→2 g (x)
x→2 2x
2·2
4
lim
x→2
As x → 2, f (x) = x − 2 → 0 and g(x) = x2 − 4 → 0, but the quotient
f (x)
x−2
0
= 2
→ .
g(x)
x −4
0
We can not determine a value for “ 00 ” without some extra work!
f (x)
might equal any number or even fail
g(x)
to exist! [1ex] Condensed: The “form” 00 is indeterminate.
If lim f (x) = lim g(x) = 0, then lim
x→a
1.2
x→a
x→a
Other Indeterminate Forms
Indeterminate Forms
Indeterminate Forms
• The most basic indeterminate form is 00 .
f (x)
x→a g(x)
• It is indeterminate because, if lim f (x) = lim g(x) = 0, then lim
x→a
x→a
might equal any number or even fail to exist!
sin x
e−x
, lim
, ···.
x→∞ sin(1/x)
x→0 x
• Specific cases: lim
• The forms
∞
∞
and 0 · ∞ are also indeterminate.
1
• They are actually equivalent to 00 , since any specific case of one can be
recast as a specific case of another.
• For example,
x
ln x
= lim
1/x x→0+ 1/ ln x
∞
0
∞
0
lim x ln x = lim
x→0+
x→0+
0·∞
Quiz
Quiz
n
n+2
∞
n
1+
n+2
∞
1−
1. Give the limit for
(a) 0,
2. Give the limit for
(a) 0,
2
2.1
n=1
(b) 1, (c) 2.
n=1
(b) 1, (c) 2.
L’Hôpital’s Rule
L’Hôpital’s Rule
L’Hôpital’s Rule
L’Hôpital’s Rule
f (x)
0
∞
f (x)
f 0 (x)
Suppose that lim
is a or
. Then lim
= lim 0
x→? g(x)
x→? g(x)
x→? g (x)
0
∞
Examples 2.
0
1
x−2
1
1
= lim
lim 2
=
=
x→2 x − 4
x→2 2x
0
2·2
4
sin x
lim
x→0 x
0
cos x
1
= lim
= =1
x→0
0
1
1
x − sin x
lim
x→0 x sin x
0
1 − cos x
0
= lim
still
x→0 sin x + x cos x
0
0
= lim
x→0
0
sin x
= =0
cos x + cos x − x sin x
2
2
What is Wrong with This?
What is wrong with the
following argument?
x2
0
2x
2
2
lim
= lim
= lim
=
= −∞
0
−0
x→0+ sin x
x→0+ cos x
x→0+ − sin x
Remark
L’Hôpital’s rule does not apply in cases where the numerator or the denominator
has a finite non-zero limit!!! For example,
2x
0
lim
= = 0,
1
x→0+ cos x
but a blind application of L’Hôpital’s rule leads incorrectly to
2
2x
2
= lim+
=
= −∞
lim+
cos
x
−
sin
x
−0
x→0
x→0
Quiz
Quiz
3. Give the limit for
n 1 o∞
nn
n=1
(a) 1,
(b) 2, (c) e.
4. Give the limit for
(a) 1,
3
3.1
1+
1
n
n ∞
n=1
(b) 2, (c) e.
More Examples
Form
0
0
Form 00 3.
Examples
0
cos x − 1 0
= lim
still!
x→0
0
3x2
0
− sin x 0
− cos x
−1
= lim
still! = lim
=
x→0
x→0
6x
0
6
6
sin x − x
lim
x→0
x3
2 cos x − 2 + x2
x→0
x4
lim
−2 sin x + 2x 0
0
= lim
still!
x→0
0
4x3
0
−2 cos x + 2 0
still!
= lim
x→0
12x2
0
2 sin x 0
2 cos x
2
1
= lim
still! = lim
=
=
x→0 24x
x→0
0
24
24
12
3
3.2
Form
∞
∞
∞
Form ∞
Examples
4.
2x ∞ x2 ∞ =
lim
still!
x→∞ ex
x→∞ ex
∞
∞
lim
= lim
x→∞
2
2
=
=0
ex
∞
3x ln 2 + 2x ln 2
ln(3x + 2x ) ∞ = lim
x→∞
x→∞
x
∞
3x + 2x
lim
ln 3 + 0
ln 3 + (2/3)x ln 2
=
= ln 3
x→∞
1 + (2/3)x
1+0
= lim
3.3
Limit Properties of ln x and ex
Limit Properties of ln x
lim xα ln x = 0,
x→0+
α > 0.
Proof.
ln x ∞ ∞
x→0 x−α
xα
1/x
= lim
=0
= lim+
x→0+ −α
x→0 −αx−α−1
lim+ xα ln x (0 · ∞) = lim+
x→0
lim
x→∞
ln x
= 0,
xα
α > 0.
Proof.
ln x ∞ 1
1/x
= lim
= lim
=0
x→∞ xα
x→∞ αxα−1
x→∞ αxα−1
∞
lim
Remark
• ln x tends to infinity slower than any positive power of x.
• Any positive power of ln x tends to infinity slower than x.
4
Limit Properties of ex
xn
= 0.
x→∞ ex
lim
Proof.
xn ∞ nxn−1 ∞ =
lim
x→∞ ex
x→∞
∞
ex
∞
n−2 ∞
n(n − 1)x
= lim
x→∞
ex
∞
lim
= ···
= lim
x→∞
n!
n!
=
=0
ex
∞
Remark
ex tends to infinity fasterr than any positive power of x.
3.4
Exponential Forms: 1∞ , 00 , and ∞0
Exponential Forms: 1∞ , 00 , and ∞0
Since ln xp = p ln x, the log can be used to convert each of exponential forms
1∞ , 00 , and ∞0 to 0 · ∞:
ln(1∞ ) = ∞ · 0, ln(00 ) = ∞ · 0, ln(∞0 ) = 0 · ∞.
If lim ln f (x) = L, then lim f (x) = eL .
x→?
x→?
Example 5.
lim+ xx 00 = e0 = 1
x→0
lim ln(xx ) ln 00
x→0+
= lim+ x ln x (0 · ∞)
x→0
1/x
ln x ∞ = lim
= lim (−x) = 0
= lim
∞
x→0+ 1/x
x→0+
x→0+ −1/x2
Examples
6.
More Exponential
Forms: 1∞ , 00 , and ∞0
lim x1/x ∞0 = e0 = 1
x→∞
lim (1 + x)1/x (1∞ ) = e1 = e
x→0+
5
ln x ∞ 1/x
= lim
=0
x→∞ x
x→∞
x→∞ 1
∞
ln(1 + x) 0
1/x
∞
lim ln(1 + x)
(ln 1 ) = lim
x
0
x→0+
x→0+
1
= lim
=1
x→0+ 1 + x
ln ∞0
lim ln x1/x
= lim
1
• Set x = n, we have the familiar result: as n → ∞, n n → 1.
n
• Set x = 1/n, we have: as n → ∞, 1 + n1 → e.
3.5
Form ∞ − ∞
Form ∞ − ∞
The form ∞ − ∞ can be handled by converting it to a quotient.
lim
x→(π/2)
(tan x − sec x) (∞ − ∞) =
−
=
lim
x→∞
lim
x→(π/2)−
x+e−x
e
x
−e
= lim
x→∞
x→(π/2)−
sin x − 1
cos x
0
0
cos x
0
=
=0
− sin x
−1
(∞ − ∞) = lim
x→∞
ee
lim
ee
−x
−1
e−x
0
0
−x
−x
(−e−x )
= lim ee = e0 = 1
−x
x→∞
−e
sin x
1
sin x − 1
−
=
cos x cos x
cos x
−x
−x
ee − 1
− ex = ex ee − 1 =
e−x
tan x − sec x =
ex+e
−x
Outline
Contents
1 Indeterminate Forms
1.1 (0/0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 (∞/∞) and Others . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1
1
2 L’Hôpital’s Rule
2.1 Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
2
6
3 Examples
3.1 00 . . . . . .
3.2 ∞
∞ . . . . .
3.3 ln x and ex
3.4 Exponential
3.5 ∞ − ∞ . .
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Forms
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