L'Hôpital's Rule allows us to resolve limits of indeterminate form: 0/0, ∞/∞, ∞-∞, 0^0, 1^∞, and ∞^0

1. . V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule
. Sec on .
Notes
Sec on 3.7
Indeterminate forms and lHôpital’s
Rule
V63.0121.001: Calculus I
Professor Ma hew Leingang
New York University
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Notes
Announcements
Midterm has been
returned. Please see FAQ
on Blackboard (under
”Exams and Quizzes”)
Quiz 3 this week in
recita on on Sec on 2.6,
2.8, 3.1, 3.2
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Notes
Objectives
Know when a limit is of
indeterminate form:
indeterminate quo ents:
0/0, ∞/∞
indeterminate products:
0×∞
indeterminate
differences: ∞ − ∞
indeterminate powers:
00 , ∞0 , and 1∞
Resolve limits in
indeterminate form
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2. . V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule
. Sec on .
Notes
Recall
Recall the limit laws from Chapter 2.
Limit of a sum is the sum of the limits
Limit of a difference is the difference of the limits
Limit of a product is the product of the limits
Limit of a quo ent is the quo ent of the limits ... whoops! This
is true as long as you don’t try to divide by zero.
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Notes
More about dividing limits
We know dividing by zero is bad.
Most of the me, if an expression’s numerator approaches a
finite nonzero number and denominator approaches zero, the
quo ent has an infinite. For example:
1 cos x
lim+ = +∞ lim− = −∞
x→0 x x→0 x3
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Notes
Why 1/0 ̸= ∞
1
Consider the func on f(x) = 1 .
x sin x
y
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x
Then lim f(x) is of the form 1/0, but the limit does not exist and is
x→∞
not infinite.
Even less predictable: when numerator and denominator both go to
zero.
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3. . V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule
. Sec on .
Notes
Experiments with funny limits
sin2 x
lim =0
x→0 x
x
lim does not exist
x→0 sin2 x
2
sin x .
lim 2)
=1
x→0 sin(x
sin 3x
lim =3
x→0 sin x
0
All of these are of the form , and since we can get different
0
answers in different cases, we say this form is indeterminate.
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Notes
Experiments with funny limits
sin2 x
lim =0
x→0 x
x
lim does not exist
x→0 sin2 x
2
sin x .
lim =1
x→0 sin(x2 )
sin 3x
lim =3
x→0 sin x
0
All of these are of the form , and since we can get different
0
answers in different cases, we say this form is indeterminate.
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Notes
Experiments with funny limits
sin2 x
lim =0
x→0 x
x
lim does not exist
x→0 sin2 x
2
sin x .
lim 2)
=1
x→0 sin(x
sin 3x
lim =3
x→0 sin x
0
All of these are of the form , and since we can get different
0
answers in different cases, we say this form is indeterminate.
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4. . V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule
. Sec on .
Notes
Experiments with funny limits
sin2 x
lim =0
x→0 x
x
lim does not exist
x→0 sin2 x
2
sin x .
lim 2)
=1
x→0 sin(x
sin 3x
lim =3
x→0 sin x
0
All of these are of the form , and since we can get different
0
answers in different cases, we say this form is indeterminate.
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Language Note Notes
It depends on what the meaning of the word “is” is
Be careful with the language here. We
are not saying that the limit in each
0
case “is” , and therefore nonexistent
0
because this expression is undefined.
0
The limit is of the form , which means
0
we cannot evaluate it with our limit
laws.
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Notes
Indeterminate forms are like Tug Of War
. Which side wins depends on which side is stronger.
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5. . V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule
. Sec on .
Notes
Outline
L’Hôpital’s Rule
Rela ve Rates of Growth
Other Indeterminate Limits
Indeterminate Products
Indeterminate Differences
Indeterminate Powers
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Notes
The Linear Case
Ques on
f(x)
If f and g are lines and f(a) = g(a) = 0, what is lim ?
x→a g(x)
Solu on
The func ons f and g can be wri en in the form
f(x) = m1 (x − a) g(x) = m2 (x − a)
So
f(x) m1
=
g(x) m2
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Notes
The Linear Case, Illustrated
y y = g(x)
y = f(x)
a f(x) g(x)
. x
x
f(x) f(x) − f(a) (f(x) − f(a))/(x − a) m1
= = =
g(x) g(x) − g(a) (g(x) − g(a))/(x − a) m2
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6. . V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule
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Notes
What then?
But what if the func ons aren’t linear?
Can we approximate a func on near a point with a linear
func on?
What would be the slope of that linear func on?
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Notes
Theorem of the Day
Theorem (L’Hopital’s Rule)
Suppose f and g are differen able func ons and g′ (x) ̸= 0 near a
(except possibly at a). Suppose that
lim f(x) = 0 and lim g(x) = 0
x→a x→a
or lim f(x) = ±∞ and lim g(x) = ±∞
x→a x→a
Then
f(x) f′ (x)
lim = lim ′ ,
x→a g(x) x→a g (x)
if the limit on the right-hand side is finite, ∞, or −∞.
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Notes
Meet the Mathematician
wanted to be a military man, but
poor eyesight forced him into
math
did some math on his own
(solved the “brachistocrone
problem”)
paid a s pend to Johann
Bernoulli, who proved this Guillaume François Antoine,
theorem and named it a er him! Marquis de L’Hôpital
(French, 1661–1704)
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7. . V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule
. Sec on .
Notes
Revisiting the previous examples
Example
sin2 x H 2 sin x cos x
lim = lim =0
x→0 x x→0 1
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Notes
Revisiting the previous examples
Example
sin2 x H sin x cos x H
2 cos2 x − sin2 x
lim = lim = lim =1
x→0 sin x 2 x→0 (cos x 2 ) (2x)
x→0 cos x2 − 2x2 sin(x2 )
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Notes
Revisiting the previous examples
Example
sin 3x H 3 cos 3x
lim = lim = 3.
x→0 sin x x→0 cos x
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8. . V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule
. Sec on .
Notes
Beware of Red Herrings
Example
Find
x
lim
x→0 cos x
Solu on
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Notes
Outline
L’Hôpital’s Rule
Rela ve Rates of Growth
Other Indeterminate Limits
Indeterminate Products
Indeterminate Differences
Indeterminate Powers
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Limits of Rational Functions Notes
revisited
Example
5x2 + 3x − 1
Find lim if it exists.
x→∞ 3x2 + 7x + 27
Solu on
Using L’Hôpital:
5x2 + 3x − 1 H 10x + 3 H 10 5
lim = lim = lim =
x→∞ 3x2 + 7x + 27 x→∞ 6x + 7 x→∞ 6 3
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9. . V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule
. Sec on .
Limits of Rational Functions Notes
revisited II
Example
5x2 + 3x − 1
Find lim if it exists.
x→∞ 7x + 27
Solu on
Using L’Hôpital:
5x2 + 3x − 1 H 10x + 3
lim = lim =∞
x→∞ 7x + 27 x→∞ 7
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Limits of Rational Functions Notes
revisited III
Example
4x + 7
Find lim if it exists.
x→∞ 3x2 + 7x + 27
Solu on
Using L’Hôpital:
4x + 7 H 4
lim = lim =0
x→∞ 3x2 + 7x + 27 x→∞ 6x + 7
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Notes
Limits of Rational Functions
Fact
Let f(x) and g(x) be polynomials of degree p and q.
f(x)
If p q, then lim =∞
x→∞ g(x)
f(x)
If p q, then lim =0
x→∞ g(x)
f(x)
If p = q, then lim is the ra o of the leading coefficients of
x→∞ g(x)
f and g.
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10. . V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule
. Sec on .
Notes
Exponential vs. geometric growth
Example
ex
Find lim , if it exists.
x→∞ x2
Solu on
We have
ex H ex H ex
lim = lim = lim = ∞.
x→∞ x2 x→∞ 2x x→∞ 2
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Notes
Exponential vs. geometric growth
Example
ex
What about lim ?
x→∞ x3
Answer
Solu on
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Notes
Exponential vs. fractional powers
Example
ex
Find lim √ , if it exists.
x→∞ x
Solu on (without L’Hôpital)
We have for all x 1, x1/2 x1 , so
ex ex
1/2
x x
The right hand side tends to ∞, so the le -hand side must, too.
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11. . V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule
. Sec on .
Notes
Exponential vs. fractional powers
Example
ex
Find lim √ , if it exists.
x→∞ x
Solu on (with L’Hôpital)
ex ex √
lim √ = lim 1 −1/2 = lim 2 xex = ∞
x→∞ x x→∞ x
2
x→∞
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Notes
Exponential vs. any power
Theorem
ex
Let r be any posi ve number. Then lim = ∞.
x→∞ xr
Proof.
If r is a posi ve integer, then apply L’Hôpital’s rule r mes to the frac-
on. You get
ex H H ex
lim = . . . = lim = ∞.
x→∞ xr x→∞ r!
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Notes
Exponential vs. any power
Theorem
ex
Let r be any posi ve number. Then lim = ∞.
x→∞ xr
Proof.
If r is not an integer, let m be the smallest integer greater than r. Then
ex ex
if x 1, xr xm , so r m . The right-hand side tends to ∞ by the
x x
previous step.
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12. . V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule
. Sec on .
Notes
Any exponential vs. any power
Theorem
ax
Let a 1 and r 0. Then lim = ∞.
x→∞ xr
Proof.
If r is a posi ve integer, we have
ax H H (ln a)r ax
lim = . . . = lim = ∞.
x→∞ xr x→∞ r!
If r isn’t an integer, we can compare it as before.
(1.00000001)x
So even lim = ∞!
. x→∞ x100000000
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Notes
Logarithmic versus power growth
Theorem
ln x
Let r be any posi ve number. Then lim = 0.
x→∞ xr
Proof.
One applica on of L’Hôpital’s Rule here suffices:
ln x H 1/x 1
lim = lim r−1 = lim r = 0.
x→∞ xr x→∞ rx x→∞ rx
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Notes
Outline
L’Hôpital’s Rule
Rela ve Rates of Growth
Other Indeterminate Limits
Indeterminate Products
Indeterminate Differences
Indeterminate Powers
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13. . V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule
. Sec on .
Notes
Indeterminate products
Example
√
Find lim+ x ln x
x→0
This limit is of the form 0 · (−∞).
Solu on
Jury-rig the expression to make an indeterminate quo ent. Then
apply L’Hôpital’s Rule:
√ ln x H x−1 √
lim x ln x = lim+ 1 √ = lim+ 1 −3/2 = lim+ −2 x = 0
x→0+ x→0 / x x→0 − 2 x x→0
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Notes
Indeterminate differences
Example
( )
1
lim − cot 2x
x→0+ x
This limit is of the form ∞ − ∞.
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Notes
Indeterminate Differences
Solu on
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14. . V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule
. Sec on .
Notes
Indeterminate powers
Example
Find lim+ (1 − 2x)1/x
x→0
Solu on
Take the logarithm:
( ) ( ) ln(1 − 2x)
ln lim+ (1 − 2x)1/x = lim+ ln (1 − 2x)1/x = lim+
x→0 x→0 x→0 x
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Notes
Indeterminate powers
Example
Find lim+ (1 − 2x)1/x
x→0
Solu on
0
This limit is of the form , so we can use L’Hôpital:
0
−2
ln(1 − 2x) H
lim+ = lim+ 1−2x = −2
x→0 x x→0 1
This is not the answer, it’s the log of the answer! So the answer we
. want is e−2 .
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Notes
Another indeterminate power limit
Example
Find lim (3x)4x
x→0
Solu on
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15. . V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule
. Sec on .
Notes
Summary
Form Method
0
0 L’Hôpital’s rule directly
∞
∞ L’Hôpital’s rule directly
∞
0·∞ jiggle to make 0 or
0 ∞.
∞ − ∞ combine to make an indeterminate product or quo ent
00 take ln to make an indeterminate product
∞0 di o
1∞ di o
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Notes
Final Thoughts
L’Hôpital’s Rule only works on indeterminate quo ents
Luckily, most indeterminate limits can be transformed into
indeterminate quo ents
L’Hôpital’s Rule gives wrong answers for non-indeterminate
limits!
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Notes
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