Ak máme funkcie
f
(
x
)
,
g
(
x
)
{\displaystyle f(x),g(x)}
c platí
lim
x
→
c
f
(
x
)
=
0
{\displaystyle \lim _{x\to c}f(x)=0}
lim
x
→
c
g
(
x
)
=
0
{\displaystyle \lim _{x\to c}g(x)=0}
limita
lim
x
→
c
f
′
(
x
)
g
′
(
x
)
{\displaystyle \lim _{x\to c}{\frac {f^{\prime }(x)}{g^{\prime }(x)}}}
lim
x
→
c
f
(
x
)
g
(
x
)
=
lim
x
→
c
f
′
(
x
)
g
′
(
x
)
{\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f^{\prime }(x)}{g^{\prime }(x)}}}
kde
′
{\displaystyle {}^{\prime }}
deriváciu funkcie.
Podobne v prípade, kedy máme funkcie
f
(
x
)
,
g
(
x
)
{\displaystyle f(x),g(x)}
c platí
lim
x
→
c
f
(
x
)
=
+
−
∞
{\displaystyle \lim _{x\to c}f(x)=+-\infty }
lim
x
→
c
g
(
x
)
=
+
−
∞
{\displaystyle \lim _{x\to c}g(x)=+-\infty }
lim
x
→
c
f
′
(
x
)
g
′
(
x
)
{\displaystyle \lim _{x\to c}{\frac {f^{\prime }(x)}{g^{\prime }(x)}}}
lim
x
→
c
f
(
x
)
g
(
x
)
=
lim
x
→
c
f
′
(
x
)
g
′
(
x
)
{\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f^{\prime }(x)}{g^{\prime }(x)}}}
Uvedené l'Hospitalove pravidlá sú použiteľné aj v nevlastných bodoch .
Ak je
f
′
(
x
)
g
′
(
x
)
{\displaystyle {\frac {f^{\prime }(x)}{g^{\prime }(x)}}}
c opäť neurčitým výrazom, možno l’Hospitalove pravidlá použiť opakovane. Takto môžeme postupovať, dokiaľ nezískame nejaký výraz, ktorý nie je neurčitý.
Úprava výrazov pre použitie l’Hospitalovho pravidla
upraviť
l’Hospitalove pravidlá sú definované len pre neurčité výrazy typu
0
0
{\displaystyle {\frac {0}{0}}}
∞
∞
{\displaystyle {\frac {\infty }{\infty }}}
Uvažujme ďalej funkcie
f
(
x
)
,
g
(
x
)
{\displaystyle f(x),g(x)}
c naberajú hodnôt 0 alebo
∞
{\displaystyle \infty }
Ak
f
(
x
)
⋅
g
(
x
)
{\displaystyle f(x)\cdot g(x)}
c výraz
0
⋅
∞
{\displaystyle 0\cdot \infty }
f
(
x
)
1
g
(
x
)
{\displaystyle {\frac {f(x)}{\frac {1}{g(x)}}}}
0
0
{\displaystyle {\frac {0}{0}}}
g
(
x
)
1
f
(
x
)
{\displaystyle {\frac {g(x)}{\frac {1}{f(x)}}}}
∞
∞
{\displaystyle {\frac {\infty }{\infty }}}
Ak
f
(
x
)
−
g
(
x
)
{\displaystyle f(x)-g(x)}
c výraz typu
∞
−
∞
{\displaystyle \infty -\infty }
1
g
(
x
)
−
1
f
(
x
)
1
f
(
x
)
⋅
g
(
x
)
{\displaystyle {\frac {{\frac {1}{g(x)}}-{\frac {1}{f(x)}}}{\frac {1}{f(x)\cdot g(x)}}}}
0
0
{\displaystyle {\frac {0}{0}}}
Ak
f
(
x
)
g
(
x
)
{\displaystyle {f(x)}^{g(x)}}
c výraz typu
0
0
{\displaystyle 0^{0}}
f
(
x
)
g
(
x
)
=
e
g
(
x
)
⋅
ln
f
(
x
)
{\displaystyle {f(x)}^{g(x)}=\mathrm {e} ^{g(x)\cdot \ln f(x)}}
0
⋅
∞
{\displaystyle 0\cdot \infty }
0
0
{\displaystyle {\frac {0}{0}}}
∞
∞
{\displaystyle {\frac {\infty }{\infty }}}
lim
x
→
c
e
g
(
c
)
⋅
ln
f
(
x
)
=
e
lim
x
→
c
[
g
(
x
)
⋅
ln
f
(
c
)
]
{\displaystyle \lim _{x\to c}\mathrm {e} ^{g(c)\cdot \ln f(x)}=\mathrm {e} ^{\lim _{x\to c}[g(x)\cdot \ln f(c)]}}
Ak
f
(
x
)
g
(
x
)
{\displaystyle {f(x)}^{g(x)}}
c výraz typu
∞
0
{\displaystyle \infty ^{0}}
f
(
x
)
g
(
x
)
=
e
g
(
x
)
⋅
ln
f
(
x
)
{\displaystyle {f(x)}^{g(x)}=\mathrm {e} ^{g(x)\cdot \ln f(x)}}
0
⋅
∞
{\displaystyle 0\cdot \infty }
Ak
f
(
x
)
g
(
x
)
{\displaystyle {f(x)}^{g(x)}}
c výraz typu
1
∞
{\displaystyle 1^{\infty }}
f
(
x
)
g
(
x
)
=
e
g
(
x
)
⋅
ln
f
(
x
)
{\displaystyle {f(x)}^{g(x)}=\mathrm {e} ^{g(x)\cdot \ln f(x)}}
∞
⋅
0
{\displaystyle \infty \cdot 0}
Výraz
ln
x
x
3
{\displaystyle {\frac {\ln x}{x^{3}}}}
x
→
+
∞
{\displaystyle x\to +\infty }
∞
∞
{\displaystyle {\frac {\infty }{\infty }}}
lim
x
→
+
∞
ln
x
x
3
=
lim
x
→
+
∞
1
x
3
x
2
=
lim
x
→
+
∞
1
3
x
3
=
lim
x
→
+
∞
1
∞
=
0
{\displaystyle \lim _{x\to +\infty }{\frac {\ln x}{x^{3}}}=\lim _{x\to +\infty }{\frac {\frac {1}{x}}{3x^{2}}}=\lim _{x\to +\infty }{\frac {1}{3x^{3}}}=\lim _{x\to +\infty }{\frac {1}{\infty }}=0}
Neurčitý výraz typu
(
0
⋅
∞
)
{\displaystyle (0\cdot \infty )}
f(x)g(x) na podiel
f
(
x
)
1
g
(
x
)
{\displaystyle {\frac {f(x)}{\frac {1}{g(x)}}}}
g
(
x
)
1
f
(
x
)
{\displaystyle {\frac {g(x)}{\frac {1}{f(x)}}}}
0
0
{\displaystyle {\frac {0}{0}}}
∞
∞
{\displaystyle {\frac {\infty }{\infty }}}
lim
x
→
0
+
(
x
⋅
ln
x
)
=
lim
x
→
0
+
ln
x
1
x
=
lim
x
→
0
+
(
ln
x
)
′
(
x
−
1
)
′
=
lim
x
→
0
+
1
x
−
x
−
2
=
−
lim
x
→
0
+
1
x
1
x
2
=
−
lim
x
→
0
+
x
=
0
{\displaystyle \lim _{x\to 0_{+}}(x\cdot \ln x)=\lim _{x\to 0_{+}}{\frac {\ln x}{\frac {1}{x}}}=\lim _{x\to 0_{+}}{\frac {(\ln x)^{'}}{(x^{-1})'}}=\lim _{x\to 0_{+}}{\frac {\frac {1}{x}}{-x^{-2}}}=-\lim _{x\to 0_{+}}{\frac {\frac {1}{x}}{\frac {1}{x^{2}}}}=-\lim _{x\to 0_{+}}x=0}
Externé odkazy
upraviť
![{\displaystyle \lim _{x\to c}\mathrm {e} ^{g(c)\cdot \ln f(x)}=\mathrm {e} ^{\lim _{x\to c}[g(x)\cdot \ln f(c)]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/480b3269a95637c1f723e29ae83f96a7b3e64fcf)
