MATHEMATICS B1
Barbora Batíková
VŠFS Praha
[email protected]
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1
6. Differentiation of Functions of One Variable
6.2. l’Hospital’s rule
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6. Differentiation of Functions of One Variable
1
6.2. l’Hospital’s rule
6. Differentiation of Functions of One Variable
6.2. l’Hospital’s rule
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6. Differentiation of Functions of One Variable
6.2. l’Hospital’s rule
Theorem
(6.2. L’Hospital’s rule).
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6. Differentiation of Functions of One Variable
Theorem
(6.2. L’Hospital’s rule).
If the limit
lim
x→c
6.2. l’Hospital’s rule
f (x)
g(x)
is of the type
”
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0
±∞
or
”,
0
±∞
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6. Differentiation of Functions of One Variable
Theorem
(6.2. L’Hospital’s rule).
If the limit
lim
x→c
6.2. l’Hospital’s rule
f (x)
g(x)
is of the type
”
0
±∞
or
”,
0
±∞
then
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6. Differentiation of Functions of One Variable
Theorem
(6.2. L’Hospital’s rule).
If the limit
lim
x→c
6.2. l’Hospital’s rule
f (x)
g(x)
is of the type
”
0
±∞
or
”,
0
±∞
then
lim
x→c
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f (x)
f 0 (x)
= lim 0
,
g(x) x→c g (x)
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6. Differentiation of Functions of One Variable
Theorem
(6.2. L’Hospital’s rule).
If the limit
lim
x→c
6.2. l’Hospital’s rule
f (x)
g(x)
is of the type
”
0
±∞
or
”,
0
±∞
then
lim
x→c
f (x)
f 0 (x)
= lim 0
,
g(x) x→c g (x)
whenever the expression on the right hand side exists.
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6. Differentiation of Functions of One Variable
6.2. l’Hospital’s rule
Example
1) lim
x→0
sin x
x ,
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6. Differentiation of Functions of One Variable
6.2. l’Hospital’s rule
Example
sin x
x ,
lim arcsin 2x ,
x→0 tan 3x
1) lim
x→0
2)
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6. Differentiation of Functions of One Variable
6.2. l’Hospital’s rule
Example
sin x
x ,
lim arcsin 2x ,
x→0 tan 3x
3x
,
lim 1−cos
x2
x→0
1) lim
x→0
2)
3)
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6. Differentiation of Functions of One Variable
6.2. l’Hospital’s rule
Example
sin x
x ,
lim arcsin 2x ,
x→0 tan 3x
3x
,
lim 1−cos
x2
x→0
3x
lim e 2 ,
x→∞ x
1) lim
x→0
2)
3)
4)
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6. Differentiation of Functions of One Variable
6.2. l’Hospital’s rule
Example
sin x
x ,
lim arcsin 2x ,
x→0 tan 3x
3x
,
lim 1−cos
x2
x→0
3x
lim e 2 ,
x→∞ x
1) lim
x→0
2)
3)
4)
5) lim x cot 2x,
x→0+
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6. Differentiation of Functions of One Variable
6.2. l’Hospital’s rule
Example
sin x
x ,
lim arcsin 2x ,
x→0 tan 3x
3x
,
lim 1−cos
x2
x→0
3x
lim e 2 ,
x→∞ x
1) lim
x→0
2)
3)
4)
5) lim x cot 2x,
x→0+
√
6) lim x ln x,
x→0+
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6. Differentiation of Functions of One Variable
6.2. l’Hospital’s rule
Example
sin x
x ,
lim arcsin 2x ,
x→0 tan 3x
3x
,
lim 1−cos
x2
x→0
3x
lim e 2 ,
x→∞ x
1) lim
x→0
2)
3)
4)
5) lim x cot 2x,
x→0+
√
6) lim x ln x,
x→0+
7) lim
x→0+
1
sin x
−
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1
x
,
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6. Differentiation of Functions of One Variable
6.2. l’Hospital’s rule
Example
sin x
x ,
lim arcsin 2x ,
x→0 tan 3x
3x
,
lim 1−cos
x2
x→0
3x
lim e 2 ,
x→∞ x
1) lim
x→0
2)
3)
4)
5) lim x cot 2x,
x→0+
√
6) lim x ln x,
x→0+
7) lim
x→0+
1
sin x
1
x ,
ex ),
−
8) lim (ln x −
x→∞
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6. Differentiation of Functions of One Variable
6.2. l’Hospital’s rule
Example
sin x
x ,
lim arcsin 2x ,
x→0 tan 3x
3x
,
lim 1−cos
x2
x→0
3x
lim e 2 ,
x→∞ x
1) lim
x→0
2)
3)
4)
5) lim x cot 2x,
x→0+
√
6) lim x ln x,
x→0+
7) lim
x→0+
1
sin x
1
x ,
ex ),
−
8) lim (ln x −
x→∞
9) lim x 2x .
x→0+
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