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General form

The general form of L'Hôpital's rule covers many cases. Let c and L be extended real numbers (i.e., real numbers, positive infinity, or negative infinity). Let I be an open interval containing c (for a two-sided limit) or an open interval with endpoint c (for a one-sided limit, or a limit at infinity if c is infinite). The real valued functions f and g are assumed to be differentiable on I except possibly at c, and additionally g ′ (x) ≠ 0 {\displaystyle g'(x)\neq 0} on I except possibly at c. It is also assumed that lim x → c f ′ (x) g ′ (x) = L. {\textstyle \lim \limits _{x\to c}{\frac {f'(x)}{g'(x)}}=L.} Thus, the rule applies to situations in which the ratio of the derivatives has a finite or infinite limit, but not to situations in which that ratio fluctuates permanently as x gets closer and closer to c.

If either lim x → c f (x) = lim x → c g (x) = 0 {\displaystyle \lim _{x\to c}f(x)=\lim _{x\to c}g(x)=0} or lim x → c | f (x) | = lim x → c | g (x) | = ∞, {\displaystyle \lim _{x\to c}|f(x)|=\lim _{x\to c}|g(x)|=\infty,} then lim x → c f (x) g (x) = L. {\displaystyle \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.

In the second case, the hypothesis that f diverges to infinity is not used in the proof (see note at the end of the proof section); thus, while the conditions of the rule are normally stated as above, the second sufficient condition for the rule's procedure to be valid can be more briefly stated as lim x → c | g (x) | = ∞. {\textstyle \lim _{x\to c}|g(x)|=\infty.}

The hypothesis that g ′ (x) ≠ 0 {\displaystyle g'(x)\neq 0} appears most commonly in the literature, but some authors sidestep this hypothesis by adding other hypotheses elsewhere. One method is to define the limit of a function with the additional requirement that the limiting function is defined everywhere on the relevant interval I except possibly at c. Another method is to require that both f and g be differentiable everywhere on an interval containing c.