L'Hôpital's rule

L'Hôpital's rule can be used to find the limiting form of a function.

One can also use l'Hôpital's rule to prove the following theorem.

Applying l'Hôpital's rule a single time still results in an indeterminate form.

The general form of l'Hôpital's rule covers many cases.

In this book is the first appearance of L'Hôpital's rule.

We turn to l'Hôpital's rule to find the solution for the limit:

The proof of a more general version of L'Hôpital's rule is given below.

L'Hôpital's rule can also be applied to other indeterminate forms, using first an appropriate algebraic transformation.

Sometimes l'Hôpital's rule does not lead to an answer in a finite number of steps unless a transformation of variables is applied.

This limit may be evaluated using l'Hôpital's rule:

L'Hôpital's rule is a general method for evaluating the indeterminate forms 0/0 and / .

The following table lists the indeterminate forms for the standard arithmetic operations and the transformations for applying l'Hôpital's rule.

Since the numerator and denominator are both limiting to zero, L'Hôpital's rule applies:

Repeatedly apply l'Hôpital's rule until the exponent is zero to conclude that the limit is zero.

Although l'Hôpital's rule is a powerful way of evaluating otherwise hard-to-evaluate limits, it is not always the easiest way.

For the case of the right part of the above equation is undefined so the limit needs to be taken when by invoking L'Hôpital's rule.

His name is firmly associated with l'Hôpital's rule for calculating limits involving indeterminate forms 0/0 and / .

In many cases, algebraic elimination, L'Hôpital's rule, or other methods can be used to manipulate the expression so that the limit can be evaluated.

The final limit may be evaluated using l'Hôpital's rule or by noting that it is the definition of the derivative of the sine function at zero.

Without this condition, it may be the case that and/or exhibits undampened oscillations as x approaches c. If this happens, then l'Hôpital's rule does not apply.

It is not a proof of the general l'Hôpital's rule because it is stricter in its definition, requiring both differentiability and that c be a real number.

In the limit , it can be shown using L'Hôpital's Rule that converges to the Shannon entropy:

The Stolz-Cesàro theorem can be viewed as a generalization of the Cesàro mean, but also as a l'Hôpital's rule for sequences.

Although L'Hôpital's rule applies both to 0/0 and to / , one of these may be better than the other in a particular case (because of the possibilities for algebraic simplification afterwards).

When and one is subtracted in the numerator (facilitating the use of l'Hôpital's rule), this simplifies to the case of log utility, and the income effect and substitution effect on saving exactly offset.