x → c f ( x )lim x → c g ( x )
lim x → c g ( x )
0 f(x)=\lim _{x\to c}g(x)=0} or ± ∞ , and g ′ ( x ) ≠ 0 for all x in I ∖ { c } } , and lim x → c f ′ ( x )
0 atau ± ∞ , f(x)=\lim _{x\to c}g(x)=0{\text{ atau }}\pm \infty ,}
g ′ ( x ) {\frac {f'(x)}{g'(x)}}} exists, then lim x → c f ( x )
≠ c, dan lim x → c f ′ ( x ) g ′ ( x ) {\frac {f'(x)}{g'(x)}}} ada, maka lim x → c f ( x )
However, it is believed that the rule was discovered by the Swiss mathematician Johann Bernoulli.
Meskipun demikian, aturan ini diyakini pertama kali ditemukan oleh matematikawan dari Swiss bernama Johann Bernoulli.
The general form of l'Hôpital's rule covers many cases.
Bentuk umum dari aturan L'hopital dapat digunakan untuk menyelesaikan banyak kasus.
If either a ) lim x → c f ( x )lim x → c g ( x )0 f(x)=\lim _{x\to c}g(x)=0} or b ) lim x → c | f ( x ) |lim x → c | g ( x ) |∞ , |f(x)|=\lim _{x\to c}|g(x)|=\infty ,} then lim x → c f ( x ) g ( x )lim x → c f ′ ( x ) g ′ ( x )
0 f(x)=\lim _{x\to c}g(x)=0} atau lim x → c | f ( x ) |lim x → c | g ( x ) |∞ , |f(x)|=\lim _{x\to c}|g(x)|=\infty ,} maka lim x → c f ( x )
{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f'(x)}{g'(x)}}=L.} Although we have written x → c throughout, the limits may also be one-sided limits ( x → c + } or x → c − } ), when c is a finite endpoint of I .
{\frac {f(x)}{g(x)}}=L.} Meskipun kita menulis x → c, limit di atas bisa saja berupa limit sepihak (x → c+ atau x → c−).
L {\frac {f'(x)}{g'(x)}}=L} that the function f ′ ( x )
{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f'(x)}{g'(x)}}.}
( 1 ) + 12 ≠ 0 f(x)=\lim _{x\to 1}(x+1)=(1)+1=2\neq 0} and lim x → 1 g ( x )lim x → 1 ( 2 x + 1 )
0 f(x)=\lim _{x\to c}g(x)=0} atau lim x → c | f ( x ) |lim x → c | g ( x ) |∞ , |f(x)|=\lim _{x\to c}|g(x)|=\infty ,} maka lim x → c f ( x )
{\frac {f'(x)}{g'(x)}}=\lim _{x\to 1}{\frac {(x+1)'}{(2x+1)'}}=\lim _{x\to 1}{\frac {1}{2}}={\frac {1}{2}}.}
{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f'(x)}{g'(x)}}.}
However, since f ( x ) is differentiable everywhere except c , then lim x → c f ′ ( x ) f'(x)} still exists.
≠ c, dan lim x → c f ′ ( x ) g ′ ( x ) {\frac {f'(x)}{g'(x)}}} ada, maka lim x → c f ( x )
Thus, since lim x → c f ( x ) g ( x )0 0 {\frac {f(x)}{g(x)}}={\frac {0}{0}}} and lim x → c f ′ ( x ) g ′ ( x ) {\frac {f'(x)}{g'(x)}}} exists, l'Hôpital's rule still holds.
≠ c, dan lim x → c f ′ ( x ) g ′ ( x ) {\frac {f'(x)}{g'(x)}}} ada, maka lim x → c f ( x )
g ( x ) {\frac {f(x)}{g(x)}}} may nevertheless exist.
{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f'(x)}{g'(x)}}.}
1 + 01. {\frac {f(x)}{g(x)}}=\lim _{x\to \infty }\left({\frac {x+\sin(x)}{x}}\right)=\lim _{x\to \infty }\left(1+{\frac {\sin(x)}{x}}\right)=1+0=1.}
0 f(x)=\lim _{x\to c}g(x)=0} atau lim x → c | f ( x ) |lim x → c | g ( x ) |∞ , |f(x)|=\lim _{x\to c}|g(x)|=\infty ,} maka lim x → c f ( x )
x → 0 + x ln ⁡ xlim x → 0 + ln ⁡ x 1 xH lim x → 0 + 1 x − 1 x 2lim x → 0 + − x0. }x\ln x=\lim _{x\to 0^{+}}{\frac {\ln x}{\frac
0 f(x)=\lim _{x\to c}g(x)=0} atau lim x → c | f ( x ) |lim x → c | g ( x ) |∞ , |f(x)|=\lim _{x\to c}|g(x)|=\infty ,} maka lim x → c f ( x )
− g ( c ) x − c )f ′ ( c )
g ( x )lim x → c f ′ ( x )
lim x → c f ′ ( x )
lim x → c g ( x )
{g(y)-g(x)}}={\frac {{\frac {f(y)}{g(y)}}-{\frac {f(x)}{g(y)}}}{1-{\frac {g(x)}{g(y)}}}}\leq M(x).}
{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f'(x)}{g'(x)}}.}
}{\frac {f(y)}{g(y)}}\leq \limsup _{y\in S_{x}}{\frac {f(y)}{g(y)}}\leq M(x).}
{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f'(x)}{g'(x)}}.}
m(x)=\lim _{x\to c}M(x)=\lim _{x\to c}{\frac {f'(x)}{g'(x)}}=L.} and lim x → c ( lim inf y ∈ S x f ( y )
{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f'(x)}{g'(x)}}.}
lim inf x → c f ( x )
lim x → c g ( x )
lim sup x → c f ( x )
lim x → c g ( x )
\left(\limsup _{y\in S_{x}}{\frac {f(y)}{g(y)}}\right)=\limsup _{x\to c}{\frac {f(x)}{g(x)}}.}
{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f'(x)}{g'(x)}}.}
Suppose, moreover, that lim x → a f ′ ( x ) f'(x)} exists.
≠ c, dan lim x → c f ′ ( x ) g ′ ( x ) {\frac {f'(x)}{g'(x)}}} ada, maka lim x → c f ( x )
lim x → a f ′ ( x ) .
lim x → c g ( x )
Moreover, lim x → a g ( x )
lim x → c g ( x )
Applying L'Hopital's rule shows that f ′ ( a ) :lim x → a f ( x ) − f ( a ) x − alim x → a h ′ ( x ) g ′ ( x )lim x → a f ′ ( x ) .
≠ c, dan lim x → c f ′ ( x ) g ′ ( x ) {\frac {f'(x)}{g'(x)}}} ada, maka lim x → c f ( x )
{\frac {f(x)-f(a)}{x-a}}=\lim _{x\to a}{\frac {h'(x)}{g'(x)}}=\lim _{x\to a}f'(x).}
{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f'(x)}{g'(x)}}.}