L’Hopital’s Rule is a method in calculus for finding the limit of a quotient of two functions when the limit of the numerator and denominator both approach zero or infinity. The rule states that, under certain conditions, the limit of the quotient is equal to the limit of the quotient of their derivatives:
If lim (x → a) f(x) = lim (x → a) g(x) = 0 or ±∞ and lim (x → a) f'(x) / g'(x) exists, then:
lim (x → a) f(x) / g(x) = lim (x → a) f'(x) / g'(x)
L’Hopital’s Rule can be applied repeatedly if the limit of the quotient of the derivatives is still indeterminate. It is important to note that L’Hopital’s Rule can only be used when both the numerator and denominator approach zero or infinity, and when the derivatives of the functions exist in the neighborhood of the point of interest.
L’Hopital’s Rule is useful for simplifying and evaluating limits that would otherwise be difficult or impossible to compute using basic limit properties and algebraic techniques.