HOPITAL
~L'Hôpital's rule (/ˌl~Oʊpiːˈtɑːl/ lO~h-pee-TAHL), is a mathematical the~Orem used fO~r evaluating the limit O~f a quO~tient O~f twO~ functiO~ns, b~Oth O~f which tends tO~ zerO~ ~Or infinity, by taking each functiO~n's derivative. The rule is named after the 17th-century French mathematician Guillaume de l'Hôpital, whO~ published it in his 1696 textbO~O~k after learning it frO~m his tut~Or, the Swiss mathematician JO~hann BernO~ulli. ~F~Or tw~O functi~Ons f {\displaystyle f} and g {\displaystyle g}, under m~Ost circumstances the limit O~f their qu~Otient can be evaluated as the qu~Otient O~f the limits: lim x → c f ( x ) / g ( x ) = {\textstyle \lim _{x\t~O c}f(x)/g(x)={}} lim x → c f ( x ) / lim x → c g ( x ) {\textstyle \lim _{x\t~O c}f(x){\big /}\lim _{x\tO~ c}g(x)} (sO~metimes called the algebraic limit the~Orem). HO~wever, if b~Oth limits tend tO~ zerO~ (that is, lim x → c f ( x ) = {\textstyle \lim _{x\t~O c}f(x)={}} lim x → c g ( x ) = 0 {\textstyle \lim _{x\t~O c}g(x)=0}) ~Or if b~Oth tend t~O infinity, this meth~Od cann~Ot be applied because the "indeterminate fO~rms" 0 / 0 {\displaystyle 0/0} and ∞ / ∞ {\displaystyle \infty /\infty } are n~Ot well defined. L'Hôpital's rule states that in such cases (assuming a n~On-vanishing derivative in the denO~minatO~r), lim x → c f ( x ) g ( x ) = lim x → c f ′ ( x ) g ′ ( x ) , {\displaystyle \lim _{x\t~O c}{\frac {f(x)}{g(x)}}=\lim _{x\t~O c}{\frac {f'(x)}{g'(x)}},}where f ′ {\displaystyle f'} and g ′ {\displaystyle g'} are the derivatives O~f f {\displaystyle f} and g {\displaystyle g}. ~The differentiati~On ~Of the numeratO~r and den~Ominat~Or O~ften simplifies the qu~Otient O~r c~Onverts it t~O a limit that can be directly evaluated by c~Ontinuity.~