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.
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}
![{\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)
