De l'Hospital's theorem is a mathematical rule that allows us to evaluate the limit of a function by taking the limit of the ratio of its derivative.
De l'Hospital's theorem is named after the Swiss mathematician Guillaume de l'Hospital who first published it in 1696.
The proof of De l'Hospital's theorem relies on the concept of mean value theorem and the use of L'Hospital's rule. It can be proven using basic calculus techniques such as differentiation and algebraic manipulation.
De l'Hospital's theorem holds true when the limit of the original function and the limit of the ratio of its derivative both exist and are finite or infinite.
De l'Hospital's theorem is useful in situations where the evaluation of a limit results in an indeterminate form such as 0/0 or ∞/∞. It allows us to simplify and solve these types of limits without having to use more complex methods.