L Hopital Limit.
In calculus,
L’Hôpital’s dominion
or
L’Hospital’s rule
(French:
[lopital], ,
lohpeeTAHL
), as well known every bit
Bernoulli’s dominion, is a theorem which provides a technique to evaluate limits of indeterminate forms. Application (or repeated application) of the dominion often converts an indeterminate form to an expression that can exist easily evaluated by exchange. The dominion is named after the 17thcentury French mathematician Guillaume de fifty’Hôpital. Although the dominion is often attributed to Fifty’Hospital, the theorem was outset introduced to him in 1694 by the Swiss mathematician Johann Bernoulli.
Fifty’Hôpital’due south rule states that for functions
f
and
g
which are differentiable on an open interval
I
except maybe at a point
c
contained in
I, if
and
for all
x
in
I
with
x
≠
c
, and
exists, then

$$
lim
x
→
c
f
(
x
)
g
(
ten
)
=
lim
x
→
c
f
′
(
ten
)
g
′
(
x
)
.
{\displaystyle \lim _{10\to c}{\frac {f(x)}{g(x)}}=\lim _{10\to c}{\frac {f'(x)}{1000′(ten)}}.}
The differentiation of the numerator and denominator often simplifies the caliber or converts it to a limit that can exist evaluated straight.
Daftar Isi:
 1 History [edit]
 2 Full general form [edit]
 3 Cases where theorem cannot be applied (Necessity of conditions) [edit]
 4 Examples [edit]
 5 Complications [edit]
 6 Other indeterminate forms [edit]
 7 Stolz–Cesàro theorem [edit]
 8 Geometric estimation [edit]
 9 Proof of L’Hôpital’s rule [edit]
 10 Corollary [edit]
 11 Run across also [edit]
 12 Notes [edit]
 13 References [edit]
 14 L Hopital Limit
History
[edit]
Guillaume de l’Hôpital (also written l’Hospital^{[a]}) published this rule in his 1696 book
Analyse des Infiniment Petits cascade l’Intelligence des Lignes Courbes
(literal translation:
Analysis of the Infinitely Pocketsize for the Understanding of Curved Lines), the first textbook on differential calculus.^{[i]}
^{[b]}
Notwithstanding, information technology is believed that the dominion was discovered by the Swiss mathematician Johann Bernoulli.^{[three]}
^{[4]}
Full general form
[edit]
The full general form of L’Hôpital’s rule covers many cases. Let
c
and
L
be extended real numbers (i.e., existent numbers, positive infinity, or negative infinity). Allow
I
be an open interval containing
c
(for a twosided limit) or an open interval with endpoint
c
(for a onesided limit, or a limit at infinity if
c
is infinite). The real valued functions
f
and
chiliad
are assumed to be differentiable on
I
except possibly at
c
, and additionally
on
I
except possibly at
c
. It is also assumed that
Thus the rule applies to situations in which the ratio of the derivatives has a finite or infinite limit, simply non to situations in which that ratio fluctuates permanently equally
10
gets closer and closer to
c
.
If either
$$$$
or
$$$$
then
$$$$
Although we have written
ten
→
c
throughout, the limits may also be isided limits (
x
→
c
^{+}
or
x
→
c
^{−}
), when
c
is a finite endpoint of
I
.
In the second case, the hypothesis that
f
diverges to infinity is not used in the proof (encounter note at the end of the proof section); thus, while the conditions of the rule are ordinarily stated as in a higher place, the second sufficient condition for the rule’s procedure to be valid tin be more than briefly stated as
The hypothesis that
appears most commonly in the literature, but some authors sidestep this hypothesis by adding other hypotheses elsewhere. 1 method^{[5]}
is to ascertain the limit of a function with the boosted requirement that the limiting function is defined everywhere on the relevant interval
I
except possibly at
c
.^{[c]}
Some other method^{[6]}
is to require that both
f
and
thousand
be differentiable everywhere on an interval containing
c
.
Cases where theorem cannot be applied (Necessity of conditions)
[edit]
All four weather for L’Hôpital’s rule are necessary:

Indeterminancy of form:
$$
lim
x
→
c
f
(
x
)
=
lim
x
→
c
g
(
10
)
=
{\displaystyle \lim _{x\to c}f(x)=\lim _{x\to c}g(ten)=0}
or
±
∞
{\displaystyle \pm \infty }
; and 
Differentiability of functions:
$$
f
(
10
)
{\displaystyle f(x)}
and
g
(
ten
)
{\displaystyle grand(x)}
are differentiable on an open interval
I
{\displaystyle {\mathcal {I}}}
except possibly at a point
c
{\displaystyle c}
independent in
I
{\displaystyle {\mathcal {I}}}
(the same indicate from the limit) ; and 
Nonzero derivative of denominator:
$$
g
′
(
ten
)
≠
{\displaystyle g'(x)\neq 0}
for all
x
{\displaystyle x}
in
I
{\displaystyle {\mathcal {I}}}
with
x
≠
c
{\displaystyle x\neq c}
; and 
Existence of limit of the caliber of the derivatives:
$$
lim
10
→
c
f
′
(
ten
)
g
′
(
x
)
{\displaystyle \lim _{ten\to c}{\frac {f'(ten)}{g'(x)}}}
exists.
Where ane of the to a higher place atmospheric condition is non satisfied, L’Hôpital’due south rule is not valid in general, and and then information technology cannot always be practical.
Form is non indeterminate
[edit]
The necessity of the first condition tin exist seen by considering the counterexample where the functions are
and
and the limit is
.
The kickoff status is non satisfied for this counterexample because
and
. This means that the grade is not indeterminate.
The second and third conditions are satisfied by
and
. The quaternary condition is also satisfied with
.
But, L’Hôpital’s rule fails in this counterexample, since
.
Differentiability of functions
[edit]
Differentiability of functions is a requirement because if a function is not differentiable, then the derivative of the functions is non guaranteed to be at each point in
. The fact that
is an open interval is grandfathered in from the hypothesis of the Cauchy Mean Value Theorem. The notable exception of the possibility of the functions being not differentiable at
exists because L’Hôpital’s rule but requires the derivative to exist every bit the function approaches
; the derivative does non demand to be taken at
.
For case, let
,
, and
. In this instance,
is non differentiable at
. However, since
is differentiable everywhere except
, then
still exists. Thus, since
and
exists, Fifty’Hôpital’due south dominion still holds.
Derivative of denominator is zero
[edit]
The necessity of the condition that
nigh
tin be seen by the following counterexample due to Otto Stolz.^{[7]}
Let
and
Then there is no limit for
equally
However,

$$
f
′
(
x
)
1000
′
(
x
)
=
2
cos
2
ten
(
2
cos
2
x
)
due east
sin
10
+
(
ten
+
sin
x
cos
x
)
east
sin
ten
cos
x
=
ii
cos
x
2
cos
ten
+
x
+
sin
x
cos
x
due east
−
sin
10
,
{\displaystyle {\begin{aligned}{\frac {f'(x)}{g'(x)}}&={\frac {2\cos ^{ii}x}{(2\cos ^{ii}x)e^{\sin x}+(x+\sin x\cos x)e^{\sin x}\cos 10}}\\&={\frac {2\cos ten}{ii\cos x+x+\sin x\cos 10}}e^{\sin x},\cease{aligned}}}
which tends to 0 every bit
. Further examples of this type were found by Ralph P. Boas Jr.^{[8]}
Limit of derivatives does not exist
[edit]
The requirement that the limit

$$
lim
x
→
c
f
′
(
x
)
g
′
(
10
)
{\displaystyle \lim _{x\to c}{\frac {f'(x)}{g'(x)}}}
exist is essential. Without this condition,
or
may exhibit undamped oscillations as
approaches
, in which case L’Hôpital’s dominion does not apply. For case, if
,
and
, then

$$
f
′
(
x
)
m
′
(
10
)
=
one
+
cos
(
x
)
i
;
{\displaystyle {\frac {f'(ten)}{one thousand'(x)}}={\frac {1+\cos(ten)}{1}};}
this expression does non approach a limit as
goes to
, since the cosine function oscillates between
ane
and
−ane. But working with the original functions,
can exist shown to exist:

$$
lim
x
→
∞
f
(
x
)
one thousand
(
x
)
=
lim
x
→
∞
(
x
+
sin
(
10
)
10
)
=
lim
x
→
∞
(
one
+
sin
(
x
)
x
)
=
1
+
lim
x
→
∞
(
sin
(
x
)
x
)
=
1
+
=
1.
{\displaystyle \lim _{x\to \infty }{\frac {f(x)}{g(x)}}=\lim _{x\to \infty }\left({\frac {x+\sin(x)}{10}}\right)=\lim _{ten\to \infty }\left(1+{\frac {\sin(ten)}{x}}\right)=ane+\lim _{x\to \infty }\left({\frac {\sin(x)}{ten}}\right)=i+0=i.}
In a case such as this, all that tin can be concluded is that

$$
lim inf
10
→
c
f
′
(
x
)
grand
′
(
ten
)
≤
lim inf
x
→
c
f
(
x
)
g
(
x
)
≤
lim sup
x
→
c
f
(
10
)
k
(
ten
)
≤
lim sup
x
→
c
f
′
(
x
)
chiliad
′
(
x
)
,
{\displaystyle \liminf _{x\to c}{\frac {f'(x)}{g'(x)}}\leq \liminf _{x\to c}{\frac {f(x)}{g(x)}}\leq \limsup _{ten\to c}{\frac {f(x)}{grand(x)}}\leq \limsup _{x\to c}{\frac {f'(ten)}{g'(x)}},}
so that if the limit of
f/g
exists, then it must lie between the inferior and superior limits of
f′/g′. (In the instance to a higher place, this is true, since 1 indeed lies between 0 and 2.)
Examples
[edit]
 Hither is a basic example involving the exponential part, which involves the indeterminate form
/
at
10
= 0:$$$$
lim
x
→
e
ten
−
1
x
2
+
10
=
lim
x
→
d
d
x
(
due east
x
−
1
)
d
d
x
(
x
two
+
ten
)
=
lim
x
→
east
x
ii
ten
+
one
=
1.
{\displaystyle {\begin{aligned}\lim _{x\to 0}{\frac {east^{x}1}{x^{2}+10}}&=\lim _{x\to 0}{\frac {{\frac {d}{dx}}(east^{10}ane)}{{\frac {d}{dx}}(ten^{two}+10)}}\\[4pt]&=\lim _{x\to 0}{\frac {e^{x}}{2x+1}}\\[4pt]&=1.\end{aligned}}}
 This is a more elaborate example involving
/
. Applying L’Hôpital’s rule a unmarried time however results in an indeterminate form. In this instance, the limit may exist evaluated by applying the rule iii times:$$$$
lim
x
→
2
sin
(
10
)
−
sin
(
2
ten
)
x
−
sin
(
x
)
=
lim
x
→
2
cos
(
x
)
−
2
cos
(
2
x
)
one
−
cos
(
ten
)
=
lim
ten
→
−
ii
sin
(
x
)
+
4
sin
(
2
10
)
sin
(
ten
)
=
lim
ten
→
−
2
cos
(
x
)
+
8
cos
(
2
ten
)
cos
(
10
)
=
−
2
+
8
1
=
6.
{\displaystyle {\begin{aligned}\lim _{x\to 0}{\frac {2\sin(x)\sin(2x)}{x\sin(10)}}&=\lim _{x\to 0}{\frac {ii\cos(x)2\cos(2x)}{1\cos(x)}}\\[4pt]&=\lim _{ten\to 0}{\frac {2\sin(x)+4\sin(2x)}{\sin(10)}}\\[4pt]&=\lim _{x\to 0}{\frac {2\cos(x)+8\cos(2x)}{\cos(x)}}\\[4pt]&={\frac {2+eight}{1}}\\[4pt]&=halfdozen.\terminate{aligned}}}
 Hither is an example involving
∞
/
∞
:$$$$
lim
10
→
∞
x
n
⋅
e
−
x
=
lim
x
→
∞
ten
n
due east
x
=
lim
10
→
∞
n
x
due north
−
1
e
ten
=
n
⋅
lim
x
→
∞
x
n
−
ane
e
x
.
{\displaystyle \lim _{ten\to \infty }x^{due north}\cdot e^{x}=\lim _{10\to \infty }{\frac {ten^{north}}{eastward^{10}}}=\lim _{ten\to \infty }{\frac {nx^{due north1}}{due east^{ten}}}=n\cdot \lim _{ten\to \infty }{\frac {ten^{n1}}{east^{x}}}.}
Repeatedly use L’Hôpital’southward rule until the exponent is zero (if
n
is an integer) or negative (if
n
is fractional) to conclude that the limit is zero.  Here is an example involving the indeterminate form
0 · ∞
(see below), which is rewritten as the grade
∞
/
∞
:$$$$
lim
x
→
+
x
ln
10
=
lim
ten
→
+
ln
10
i
x
=
lim
x
→
+
1
x
−
1
ten
two
=
lim
10
→
+
−
x
=
0.
{\displaystyle \lim _{x\to 0^{+}}x\ln ten=\lim _{x\to 0^{+}}{\frac {\ln x}{\frac {one}{10}}}=\lim _{x\to 0^{+}}{\frac {\frac {one}{x}}{{\frac {i}{x^{2}}}}}=\lim _{x\to 0^{+}}10=0.}
 Here is an example involving the mortgage repayment formula and
/
. Allow
P
be the master (loan amount),
r
the interest charge per unit per period and
northward
the number of periods. When
r
is cipher, the repayment corporeality per period is
$$
P
n
{\displaystyle {\frac {P}{n}}}
(since only principal is being repaid); this is consistent with the formula for nonnothing involvement rates:$$$$
lim
r
→
P
r
(
1
+
r
)
due north
(
1
+
r
)
n
−
1
=
P
lim
r
→
(
ane
+
r
)
n
+
r
n
(
ane
+
r
)
n
−
i
n
(
1
+
r
)
n
−
1
=
P
n
.
{\displaystyle {\begin{aligned}\lim _{r\to 0}{\frac {Pr(1+r)^{n}}{(ane+r)^{n}1}}&=P\lim _{r\to 0}{\frac {(1+r)^{due north}+rn(1+r)^{n1}}{due north(1+r)^{due north1}}}\\[4pt]&={\frac {P}{n}}.\end{aligned}}}
 1 can also use L’Hôpital’due south rule to prove the following theorem. If
f
is twicedifferentiable in a neighborhood of
10
and that its second derivative is continuous on this neighbourhood, and then$$$$
lim
h
→
f
(
x
+
h
)
+
f
(
x
−
h
)
−
two
f
(
ten
)
h
2
=
lim
h
→
f
′
(
ten
+
h
)
−
f
′
(
10
−
h
)
ii
h
=
lim
h
→
f
″
(
10
+
h
)
+
f
″
(
10
−
h
)
2
=
f
″
(
ten
)
.
{\displaystyle {\begin{aligned}\lim _{h\to 0}{\frac {f(x+h)+f(xh)2f(ten)}{h^{2}}}&=\lim _{h\to 0}{\frac {f'(ten+h)f'(10h)}{2h}}\\[4pt]&=\lim _{h\to 0}{\frac {f”(x+h)+f”(xh)}{2}}\\[4pt]&=f”(x).\end{aligned}}}

Sometimes L’Hôpital’s rule is invoked in a catchy style: suppose
$$
f(x) +
f′(x)
converges every bit
10
→ ∞
and that
e
x
⋅
f
(
x
)
{\displaystyle e^{10}\cdot f(10)}
converges to positive or negative infinity. Then:$$$$
lim
x
→
∞
f
(
ten
)
=
lim
ten
→
∞
eastward
x
⋅
f
(
x
)
due east
10
=
lim
x
→
∞
e
10
(
f
(
ten
)
+
f
′
(
x
)
)
east
x
=
lim
x
→
∞
(
f
(
10
)
+
f
′
(
x
)
)
{\displaystyle \lim _{x\to \infty }f(10)=\lim _{ten\to \infty }{\frac {e^{x}\cdot f(x)}{e^{x}}}=\lim _{10\to \infty }{\frac {east^{x}{\bigl (}f(x)+f'(ten){\bigr )}}{e^{ten}}}=\lim _{x\to \infty }{\bigl (}f(x)+f'(x){\bigr )}}
and so,
$$
lim
x
→
∞
f
(
x
)
{\textstyle \lim _{10\to \infty }f(x)}
exists and
lim
10
→
∞
f
′
(
x
)
=
0.
{\textstyle \lim _{x\to \infty }f'(x)=0.}
The result remains true without the added hypothesis that
$$
e
x
⋅
f
(
10
)
{\displaystyle east^{ten}\cdot f(x)}
converges to positive or negative infinity, but the justification is so incomplete.
Complications
[edit]
Sometimes L’Hôpital’s dominion does not lead to an answer in a finite number of steps unless some additional steps are applied. Examples include the following:
 Ii applications can lead to a return to the original expression that was to be evaluated:
$$$$
lim
ten
→
∞
e
x
+
eastward
−
x
east
10
−
e
−
x
=
lim
10
→
∞
e
x
−
e
−
x
e
x
+
e
−
x
=
lim
ten
→
∞
e
ten
+
e
−
x
e
10
−
e
−
ten
=
⋯
.
{\displaystyle \lim _{10\to \infty }{\frac {e^{ten}+east^{10}}{e^{ten}e^{x}}}=\lim _{ten\to \infty }{\frac {e^{10}eastward^{x}}{due east^{x}+east^{x}}}=\lim _{x\to \infty }{\frac {e^{x}+eastward^{x}}{e^{x}e^{ten}}}=\cdots .}
This situation can exist dealt with past substituting
$$
y
=
e
x
{\displaystyle y=east^{10}}
and noting that
y
goes to infinity as
x
goes to infinity; with this commutation, this problem can be solved with a single application of the rule:$$$$
lim
x
→
∞
due east
10
+
e
−
10
due east
10
−
e
−
ten
=
lim
y
→
∞
y
+
y
−
1
y
−
y
−
1
=
lim
y
→
∞
i
−
y
−
two
one
+
y
−
two
=
one
1
=
1.
{\displaystyle \lim _{x\to \infty }{\frac {due east^{x}+e^{x}}{e^{ten}e^{ten}}}=\lim _{y\to \infty }{\frac {y+y^{ane}}{yy^{1}}}=\lim _{y\to \infty }{\frac {1y^{ii}}{1+y^{two}}}={\frac {ane}{one}}=1.}
Alternatively, the numerator and denominator tin can both be multiplied by
$$
e
x
,
{\displaystyle e^{ten},}
at which signal L’Hôpital’due south rule tin can immediately be practical successfully:^{[nine]}$$$$
lim
ten
→
∞
e
x
+
e
−
x
eastward
ten
−
e
−
x
=
lim
10
→
∞
e
2
10
+
1
e
two
x
−
1
=
lim
x
→
∞
2
e
2
x
2
eastward
2
x
=
1.
{\displaystyle \lim _{ten\to \infty }{\frac {e^{10}+eastward^{x}}{e^{x}e^{ten}}}=\lim _{x\to \infty }{\frac {due east^{2x}+i}{e^{2x}1}}=\lim _{x\to \infty }{\frac {2e^{2x}}{2e^{2x}}}=ane.}
 An arbitrarily large number of applications may never atomic number 82 to an answer even without repeating:
$$$$
lim
x
→
∞
x
ane
two
+
10
−
i
ii
10
ane
2
−
ten
−
one
ii
=
lim
x
→
∞
1
2
x
−
1
two
−
ane
2
x
−
3
ii
1
2
x
−
1
ii
+
1
ii
10
−
3
2
=
lim
x
→
∞
−
ane
4
x
−
3
ii
+
iii
4
ten
−
5
2
−
1
4
10
−
3
2
−
3
four
x
−
5
two
=
⋯
.
{\displaystyle \lim _{x\to \infty }{\frac {x^{\frac {1}{2}}+x^{{\frac {1}{2}}}}{ten^{\frac {1}{2}}x^{{\frac {1}{2}}}}}=\lim _{ten\to \infty }{\frac {{\frac {1}{two}}x^{{\frac {one}{2}}}{\frac {ane}{2}}x^{{\frac {3}{2}}}}{{\frac {i}{2}}x^{{\frac {1}{2}}}+{\frac {1}{2}}x^{{\frac {3}{2}}}}}=\lim _{x\to \infty }{\frac {{\frac {1}{4}}x^{{\frac {3}{ii}}}+{\frac {3}{4}}x^{{\frac {5}{2}}}}{{\frac {1}{4}}x^{{\frac {3}{2}}}{\frac {iii}{4}}x^{{\frac {v}{2}}}}}=\cdots .}
This state of affairs too tin can be dealt with by a transformation of variables, in this instance
$$
y
=
x
{\displaystyle y={\sqrt {x}}}
:$$$$
lim
x
→
∞
x
i
ii
+
ten
−
i
2
ten
1
2
−
x
−
one
ii
=
lim
y
→
∞
y
+
y
−
i
y
−
y
−
1
=
lim
y
→
∞
1
−
y
−
two
1
+
y
−
2
=
one
1
=
1.
{\displaystyle \lim _{x\to \infty }{\frac {10^{\frac {ane}{2}}+x^{{\frac {ane}{2}}}}{x^{\frac {1}{2}}x^{{\frac {1}{2}}}}}=\lim _{y\to \infty }{\frac {y+y^{1}}{yy^{1}}}=\lim _{y\to \infty }{\frac {oney^{2}}{1+y^{two}}}={\frac {ane}{1}}=ane.}
Again, an culling approach is to multiply numerator and denominator past
$$
x
i
/
two
{\displaystyle 10^{1/2}}
before applying L’Hôpital’s rule:$$$$
lim
x
→
∞
x
1
ii
+
10
−
ane
2
x
1
two
−
x
−
1
2
=
lim
x
→
∞
x
+
1
x
−
1
=
lim
x
→
∞
1
i
=
1.
{\displaystyle \lim _{10\to \infty }{\frac {ten^{\frac {one}{2}}+x^{{\frac {1}{2}}}}{x^{\frac {i}{two}}x^{{\frac {one}{ii}}}}}=\lim _{ten\to \infty }{\frac {x+1}{ten1}}=\lim _{x\to \infty }{\frac {one}{ane}}=1.}
A common pitfall is using L’Hôpital’s rule with some circular reasoning to compute a derivative via a difference caliber. For example, consider the job of proving the derivative formula for powers of
10:

$$
lim
h
→
(
x
+
h
)
north
−
ten
north
h
=
n
x
north
−
1
.
{\displaystyle \lim _{h\to 0}{\frac {(x+h)^{n}10^{n}}{h}}=nx^{n1}.}
Applying L’Hôpital’s rule and finding the derivatives with respect to
h
of the numerator and the denominator yields
nx
^{
n−1}
as expected. Even so, differentiating the numerator requires the use of the very fact that is being proven. This is an example of begging the question, since one may not assume the fact to be proven during the course of the proof.
Other indeterminate forms
[edit]
Other indeterminate forms, such equally
1^{∞}
,
^{}
,
∞^{}
,
0 · ∞, and
∞ − ∞, tin sometimes be evaluated using L’Hôpital’s dominion. For example, to evaluate a limit involving
∞ − ∞, convert the deviation of two functions to a caliber:

$$
lim
ten
→
ane
(
10
ten
−
1
−
ane
ln
10
)
=
lim
ten
→
i
ten
⋅
ln
x
−
x
+
one
(
10
−
1
)
⋅
ln
x
(
i
)
=
lim
x
→
1
ln
x
x
−
1
10
+
ln
10
(
2
)
=
lim
x
→
1
x
⋅
ln
x
x
−
1
+
10
⋅
ln
x
(
iii
)
=
lim
x
→
1
1
+
ln
x
one
+
1
+
ln
10
(
four
)
=
lim
10
→
1
1
+
ln
10
two
+
ln
x
=
1
2
,
{\displaystyle {\begin{aligned}\lim _{x\to 1}\left({\frac {x}{xi}}{\frac {1}{\ln ten}}\correct)&=\lim _{10\to 1}{\frac {x\cdot \ln xx+one}{(x1)\cdot \ln x}}&\quad (one)\\[6pt]&=\lim _{x\to 1}{\frac {\ln 10}{{\frac {xi}{x}}+\ln x}}&\quad (2)\\[6pt]&=\lim _{ten\to ane}{\frac {x\cdot \ln x}{101+x\cdot \ln ten}}&\quad (3)\\[6pt]&=\lim _{x\to one}{\frac {1+\ln x}{1+ane+\ln ten}}&\quad (4)\\[6pt]&=\lim _{ten\to 1}{\frac {1+\ln x}{ii+\ln x}}\\[6pt]&={\frac {one}{ii}},\cease{aligned}}}
where L’Hôpital’s rule is practical when going from (ane) to (two) and again when going from (3) to (iv).
L’Hôpital’south rule can be used on indeterminate forms involving exponents by using logarithms to “move the exponent downward”. Here is an example involving the indeterminate form
^{}
:

$$
lim
x
→
+
x
x
=
lim
x
→
+
eastward
ln
(
x
ten
)
=
lim
x
→
+
e
x
⋅
ln
x
=
east
lim
x
→
+
(
10
⋅
ln
ten
)
.
{\displaystyle \lim _{ten\to 0^{+}}x^{10}=\lim _{10\to 0^{+}}due east^{\ln(ten^{x})}=\lim _{x\to 0^{+}}e^{x\cdot \ln ten}=e^{\lim \limits _{10\to 0^{+}}(10\cdot \ln 10)}.}
It is valid to motion the limit inside the exponential function considering the exponential function is continuous. Now the exponent
has been “moved downwardly”. The limit
is of the indeterminate form
0 · ∞, merely every bit shown in an example above, 50’Hôpital’s dominion may be used to make up one’s mind that

$$
lim
10
→
+
x
⋅
ln
10
=
0.
{\displaystyle \lim _{ten\to 0^{+}}ten\cdot \ln x=0.}
Thus

$$
lim
x
→
+
10
x
=
e
=
1.
{\displaystyle \lim _{ten\to 0^{+}}x^{x}=eastward^{0}=1.}
The following table lists the wellnigh common indeterminate forms, and the transformations for applying l’Hôpital’s rule:
Indeterminate form  Conditions  Transformation to $$ 

/ 
$$ 

$$ 
$$ 
$$ 
$$ 
$$ 
$$ 
$$ 
$$ 
$$ 
$$ 
$$ 
$$ 
$$ 
$$ 
$$ 
$$ 
$$ 
$$ 
Stolz–Cesàro theorem
[edit]
The Stolz–Cesàro theorem is a similar result involving limits of sequences, merely it uses finite difference operators rather than derivatives.
Geometric estimation
[edit]
Consider the curve in the aeroplane whose
x
coordinate is given by
g(t)
and whose
y
coordinate is given by
f(t), with both functions continuous, i.due east., the locus of points of the form
[grand(t),
f(t)]. Suppose
f(c) =
g(c) = 0. The limit of the ratio
f(t)
/
m(t)
every bit
t
→
c
is the slope of the tangent to the curve at the indicate
[thousand(c),
f(c)] = [0,0]. The tangent to the curve at the indicate
[chiliad(t),
f(t)]
is given by
[chiliad′(t),
f′(t)]. L’Hôpital’southward rule then states that the slope of the curve when
t
=
c
is the limit of the slope of the tangent to the curve as the curve approaches the origin, provided that this is defined.
Proof of L’Hôpital’s rule
[edit]
Special case
[edit]
The proof of L’Hôpital’s rule is elementary in the instance where
f
and
g
are continuously differentiable at the betoken
c
and where a finite limit is found after the first round of differentiation. It is not a proof of the full general L’Hôpital’s rule because information technology is stricter in its definition, requiring both differentiability and that
c
be a real number. Since many common functions have continuous derivatives (e.k. polynomials, sine and cosine, exponential functions), it is a special case worthy of attention.
Suppose that
f
and
k
are continuously differentiable at a real number
c
, that
, and that
. And then

$$
lim
x
→
c
f
(
x
)
yard
(
ten
)
=
lim
x
→
c
f
(
10
)
−
g
(
10
)
−
=
lim
x
→
c
f
(
10
)
−
f
(
c
)
g
(
x
)
−
g
(
c
)
=
lim
x
→
c
(
f
(
10
)
−
f
(
c
)
x
−
c
)
(
g
(
ten
)
−
k
(
c
)
x
−
c
)
=
lim
ten
→
c
(
f
(
ten
)
−
f
(
c
)
x
−
c
)
lim
x
→
c
(
g
(
x
)
−
grand
(
c
)
ten
−
c
)
=
f
′
(
c
)
thousand
′
(
c
)
=
lim
x
→
c
f
′
(
x
)
g
′
(
x
)
.
{\displaystyle {\begin{aligned}&\lim _{x\to c}{\frac {f(x)}{m(x)}}=\lim _{10\to c}{\frac {f(x)0}{g(ten)0}}=\lim _{10\to c}{\frac {f(ten)f(c)}{g(x)grand(c)}}\\[6pt]={}&\lim _{10\to c}{\frac {\left({\frac {f(x)f(c)}{10c}}\right)}{\left({\frac {thousand(ten)grand(c)}{xc}}\right)}}={\frac {\lim \limits _{x\to c}\left({\frac {f(x)f(c)}{xc}}\right)}{\lim \limits _{x\to c}\left({\frac {yard(ten)g(c)}{tenc}}\correct)}}={\frac {f'(c)}{g'(c)}}=\lim _{ten\to c}{\frac {f'(x)}{g'(x)}}.\finish{aligned}}}
This follows from the differencecaliber definition of the derivative. The last equality follows from the continuity of the derivatives at
c
. The limit in the decision is not indeterminate considering
.
The proof of a more general version of L’Hôpital’south rule is given below.
Full general proof
[edit]
The postobit proof is due to Taylor (1952), where a unified proof for the
/
and
±∞
/
±∞
indeterminate forms is given. Taylor notes that different proofs may be plant in Lettenmeyer (1936) and Wazewski (1949).
Let
f
and
thousand
exist functions satisfying the hypotheses in the General course section. Allow
exist the open interval in the hypothesis with endpoint
c. Considering that
on this interval and
k
is continuous,
can be chosen smaller then that
g
is nonzero on
.^{[d]}
For each
ten
in the interval, ascertain
and
as
ranges over all values between
x
and
c. (The symbols inf and sup denote the infimum and supremum.)
From the differentiability of
f
and
g
on
, Cauchy’s mean value theorem ensures that for whatever two distinct points
x
and
y
in
in that location exists a
betwixt
ten
and
y
such that
. Consequently,
for all choices of distinct
x
and
y
in the interval. The value
g(x)thousand(y) is always nonzero for singledout
x
and
y
in the interval, for if it was not, the hateful value theorem would imply the being of a
p
between
ten
and
y
such that
m’
(p)=0.
The definition of
m(x) and
Yard(x) volition result in an extended real number, and and so it is possible for them to take on the values ±∞. In the postobit two cases,
g(x) and
One thousand(10) volition establish bounds on the ratio
f
/
thou
.
Instance 1:
For any
ten
in the interval
, and indicate
y
between
ten
and
c,

$$
1000
(
ten
)
≤
f
(
x
)
−
f
(
y
)
g
(
x
)
−
g
(
y
)
=
f
(
10
)
g
(
ten
)
−
f
(
y
)
g
(
10
)
1
−
g
(
y
)
g
(
ten
)
≤
Grand
(
x
)
{\displaystyle m(x)\leq {\frac {f(10)f(y)}{g(x)one thousand(y)}}={\frac {{\frac {f(x)}{thou(ten)}}{\frac {f(y)}{g(x)}}}{1{\frac {g(y)}{yard(x)}}}}\leq Thou(10)}
and therefore as
y
approaches
c,
and
become zero, and then

$$
m
(
x
)
≤
f
(
x
)
g
(
x
)
≤
M
(
x
)
.
{\displaystyle k(x)\leq {\frac {f(10)}{one thousand(x)}}\leq M(x).}
Instance 2:
For every
x
in the interval
, ascertain
. For every point
y
betwixt
10
and
c,

$$
m
(
x
)
≤
f
(
y
)
−
f
(
ten
)
g
(
y
)
−
k
(
ten
)
=
f
(
y
)
g
(
y
)
−
f
(
x
)
g
(
y
)
1
−
1000
(
x
)
g
(
y
)
≤
M
(
x
)
.
{\displaystyle chiliad(ten)\leq {\frac {f(y)f(10)}{grand(y)g(x)}}={\frac {{\frac {f(y)}{m(y)}}{\frac {f(x)}{one thousand(y)}}}{1{\frac {one thousand(x)}{m(y)}}}}\leq M(x).}
As
y
approaches
c, both
and
become zero, and therefore

$$
1000
(
x
)
≤
lim inf
y
∈
S
x
f
(
y
)
grand
(
y
)
≤
lim sup
y
∈
Due south
x
f
(
y
)
g
(
y
)
≤
Yard
(
10
)
.
{\displaystyle m(10)\leq \liminf _{y\in S_{x}}{\frac {f(y)}{grand(y)}}\leq \limsup _{y\in S_{x}}{\frac {f(y)}{k(y)}}\leq G(ten).}
The limit superior and limit inferior are necessary since the existence of the limit of
f
/
grand
has not nevertheless been established.
It is as well the instance that

$$
lim
ten
→
c
m
(
x
)
=
lim
x
→
c
Thou
(
x
)
=
lim
x
→
c
f
′
(
x
)
g
′
(
10
)
=
50
.
{\displaystyle \lim _{ten\to c}one thousand(x)=\lim _{10\to c}Thousand(x)=\lim _{10\to c}{\frac {f'(ten)}{yard'(ten)}}=L.}
^{[e]}
and

$$
lim
ten
→
c
(
lim inf
y
∈
S
x
f
(
y
)
g
(
y
)
)
=
lim inf
ten
→
c
f
(
10
)
g
(
ten
)
{\displaystyle \lim _{x\to c}\left(\liminf _{y\in S_{x}}{\frac {f(y)}{g(y)}}\right)=\liminf _{x\to c}{\frac {f(10)}{g(x)}}}
and
lim
x
→
c
(
lim sup
y
∈
South
10
f
(
y
)
g
(
y
)
)
=
lim sup
x
→
c
f
(
x
)
k
(
x
)
.
{\displaystyle \lim _{x\to c}\left(\limsup _{y\in S_{x}}{\frac {f(y)}{thou(y)}}\correct)=\limsup _{x\to c}{\frac {f(x)}{g(x)}}.}
In instance 1, the squeeze theorem establishes that
exists and is equal to
L. In the instance 2, and the squeeze theorem again asserts that
, then the limit
exists and is equal to
L. This is the event that was to be proven.
In case 2 the assumption that
f(x) diverges to infinity was not used within the proof. This means that if g(x) diverges to infinity equally
x
approaches
c
and both
f
and
yard
satisfy the hypotheses of L’Hôpital’due south rule, then no boosted assumption is needed almost the limit of
f(10): It could even be the case that the limit of
f(x) does not exist. In this case, 50’Hopital’s theorem is really a consequence of Cesàro–Stolz.^{[10]}
In the example when g(x) diverges to infinity as
x
approaches
c
and
f(x) converges to a finite limit at
c, then L’Hôpital’s rule would be applicable, merely not absolutely necessary, since basic limit calculus will testify that the limit of
f(ten)/one thousand(x) as
x
approaches
c
must be zero.
Corollary
[edit]
A uncomplicated but very useful consequence of L’Hopital’s rule is a wellknown criterion for differentiability. It states the postobit: suppose that
f
is continuous at
a, and that
exists for all
x
in some open interval containing
a, except perhaps for
. Suppose, moreover, that
exists. And so
also exists and

$$
f
′
(
a
)
=
lim
x
→
a
f
′
(
ten
)
.
{\displaystyle f'(a)=\lim _{10\to a}f'(x).}
In particular,
f’
is also continuous at
a.
Proof
[edit]
Consider the functions
and
. The continuity of
f
at
a
tells us that
. Moreover,
since a polynomial function is always continuous everywhere. Applying L’Hopital’due south rule shows that
.
Run across also
[edit]
 L’Hôpital controversy
Notes
[edit]

^
In the 17th and 18th centuries, the name was commonly spelled “l’Hospital”, and he himself spelled his name that style. Since then, French spellings take inverse: the silent ‘due south’ has been removed and replaced with a circumflex over the preceding vowel. 
^
“Proffer I. Problême. Soit une ligne courbe AMD (AP = x, PM = y, AB = a [meet Figure 130] ) telle que la valeur de 50’appliquée y soit exprimée par une fraction, dont le numérateur & le dénominateur deviennent chacun zero lorsque x = a, c’est à dire lorsque le point P tombe sur le point donné B. On demande quelle doit être alors la valeur de l’appliquée BD. [Solution: ]…si l’on prend la difference du numérateur, & qu’on la divise par la difference du denominateur, apres avoir fait 10 = a = Ab ou AB, l’on aura la valeur cherchée de l’appliquée bd ou BD.”
Translation : “Permit there be a curve AMD (where AP = X, PM = y, AB = a) such that the value of the ordinate y is expressed by a fraction whose numerator and denominator each become cypher when ten = a; that is, when the point P falls on the given point B. Ane asks what shall then exist the value of the ordinate BD. [Solution: ]… if one takes the differential of the numerator and if one divides information technology past the differential of the denominator, after having set x = a = Ab or AB, one volition have the value [that was] sought of the ordinate bd or BD.”^{[2]}

^
The functional analysis definition of the limit of a function does not require the existence of such an interval. 
^
Since
g’
is nonzero and
g
is continuous on the interval, information technology is incommunicable for
thousand
to be aught more than once on the interval. If it had two zeros, the hateful value theorem would affirm the being of a point
p
in the interval between the zeros such that
g’
(p) = 0. And then either
g
is already nonzero on the interval, or else the interval can be reduced in size then as not to contain the single nil of
thou. 
^
The limits
$$
lim
x
→
c
m
(
x
)
{\displaystyle \lim _{x\to c}m(x)}
and
lim
x
→
c
Thousand
(
x
)
{\displaystyle \lim _{10\to c}M(x)}
both exist as they feature nondecreasing and nonincreasing functions of
x, respectively. Consider a sequence
x
i
→
c
{\displaystyle x_{i}\to c}
. And so
lim
i
m
(
10
i
)
≤
lim
i
f
′
(
ten
i
)
m
′
(
x
i
)
≤
lim
i
Thousand
(
x
i
)
{\displaystyle \lim _{i}m(x_{i})\leq \lim _{i}{\frac {f'(x_{i})}{g'(x_{i})}}\leq \lim _{i}M(x_{i})}
, as the inequality holds for each
i; this yields the inequalities
lim
x
→
c
m
(
x
)
≤
lim
10
→
c
f
′
(
x
)
g
′
(
10
)
≤
lim
x
→
c
Chiliad
(
x
)
{\displaystyle \lim _{x\to c}1000(x)\leq \lim _{x\to c}{\frac {f'(x)}{thousand'(ten)}}\leq \lim _{x\to c}Thou(10)}
The side by side footstep is to show
lim
ten
→
c
M
(
x
)
≤
lim
10
→
c
f
′
(
x
)
g
′
(
10
)
{\displaystyle \lim _{ten\to c}One thousand(x)\leq \lim _{10\to c}{\frac {f'(x)}{k'(ten)}}}
. Fix a sequence of numbers
ε
i
>
{\displaystyle \varepsilon _{i}>0}
lim
i
ε
i
=
{\displaystyle \lim _{i}\varepsilon _{i}=0}
, and a sequence
x
i
→
c
{\displaystyle x_{i}\to c}
. For each
i, choose
x
i
<
y
i
<
c
{\displaystyle x_{i}<y_{i}<c}
such that
f
′
(
y
i
)
thousand
′
(
y
i
)
+
ε
i
≥
sup
x
i
<
ξ
<
c
f
′
(
ξ
)
g
′
(
ξ
)
{\displaystyle {\frac {f'(y_{i})}{k'(y_{i})}}+\varepsilon _{i}\geq \sup _{x_{i}<\xi <c}{\frac {f'(\11 )}{g'(\xi )}}}
, by the definition of
sup
{\displaystyle \sup }
. Thus$$$$
lim
i
M
(
10
i
)
≤
lim
i
f
′
(
y
i
)
g
′
(
y
i
)
+
ε
i
=
lim
i
f
′
(
y
i
)
thousand
′
(
y
i
)
+
lim
i
ε
i
=
lim
i
f
′
(
y
i
)
g
′
(
y
i
)
{\displaystyle {\begin{aligned}\lim _{i}M(x_{i})&\leq \lim _{i}{\frac {f'(y_{i})}{thousand'(y_{i})}}+\varepsilon _{i}\\&=\lim _{i}{\frac {f'(y_{i})}{g'(y_{i})}}+\lim _{i}\varepsilon _{i}\\&=\lim _{i}{\frac {f'(y_{i})}{g'(y_{i})}}\end{aligned}}}
as desired. The argument that
$$
lim
ten
→
c
k
(
10
)
≥
lim
x
→
c
f
′
(
x
)
thousand
′
(
x
)
{\displaystyle \lim _{x\to c}m(x)\geq \lim _{ten\to c}{\frac {f'(x)}{thousand'(x)}}}
is similar.
References
[edit]

^
O’Connor, John J.; Robertson, Edmund F. “De L’Hopital biography”.
The MacTutor History of Mathematics archive. Scotland: School of Mathematics and Statistics, University of St Andrews. Retrieved
21 December
2008.

^
L’Hospital (1696).
Analyse des infiniment petits. pp. 145–146.

^
Boyer, Carl B.; Merzbach, Uta C. (2011).
A History of Mathematics
(3rd illustrated ed.). John Wiley & Sons. p. 321. ISBN978047063056iii.
Extract of page 321 
^
Weisstein, Eric W. “L’Infirmary’south Rule”.
MathWorld.

^
(Chatterjee 2005, p. 291)
harv mistake: no target: CITEREFChatterjee2005 (assist)

^
(Krantz 2004, p.79) 
^
Stolz, Otto (1879). “Ueber dice Grenzwerthe der Quotienten” [Nigh the limits of quotients].
Mathematische Annalen
(in German).
15
(3–4): 556–559. doi:10.1007/bf02086277. S2CID 122473933.

^
Boas Jr., Ralph P. (1986). “Counterexamples to L’Hopital’s Rule”.
American Mathematical Monthly.
93
(8): 644–645. doi:x.1080/00029890.1986.11971912. JSTOR 2322330.

^
Multiplying by
$$
e
−
x
{\displaystyle e^{ten}}
instead yields a solution to the limit without need for l’Hôpital’s rule. 
^
“L’Hopital’southward Theorem”.
IMOmath. International Mathematical Olympiad.
Sources
[edit]

AbhilekhChatterjee, Dipak (2005),
Existent Analysis, PHI Learning Pvt. Ltd, ISBN812032678iv

Krantz, Steven G. (2004),
A handbook of real variables. With applications to differential equations and Fourier analysis, Boston, MA: Birkhäuser Boston Inc., pp. 14+201, doi:10.1007/978081768128nine, ISBN081764329X, MR 2015447

Lettenmeyer, F. (1936), “Über die sogenannte Hospitalsche Regel”,
Journal für dice reine und angewandte Mathematik,
1936
(174): 246–247, doi:x.1515/crll.1936.174.246, S2CID 199546754

Taylor, A. E. (1952), “L’Hospital’due south rule”,
Amer. Math. Monthly,
59
(1): twenty–24, doi:10.2307/2307183, ISSN 00029890, JSTOR 2307183, MR 0044602

Wazewski, T. (1949), “Quelques démonstrations uniformes pour tous les cas du théorème de l’Hôpital. Généralisations”,
Prace Mat.Fiz.
(in French),
47: 117–128, MR 0034430
L Hopital Limit
Source: https://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule