L'Hôpital's rule

With the rule of (de) L'Hospital (pronounced [ lopi'tal ], also written L'Hôpital, or referred to as l' Hospitalsche rule or set of L'Hospital ) can be against limits of functions, which is the quotient of two can write zero convergent or divergent functions determined, calculate using the first derivatives of these functions.

The rule is named after Guillaume François Antoine, Marquis de L' Hospital ( 1661-1704 ). L'Hospital published it in 1696 in his book Analysis of infiniment petits pour l' intelligence of lignes courbes, the first textbook on differential calculus. But he had not even discovered, but taken over by Johann Bernoulli.

• 4.1 Border crossing at x0 = 0
• 4.2 Border crossing at infinity

• 5.1 subject to conditions
• 5.2 Landau - calculus

Application

The rule of L'Hospital allows in many cases to determine the limit of a function when the function term can be expressed so that upon reaching the border a vague expression arises.

All applications usually can be traced back to the task of determining the limit, when both apply, so is a vague expression of type.

The rule of L'Hospital says then that is true, if the limit on the right exists. and thereby designate the first derivatives of the function and.

The right-hand side of this equation can often be easily calculated. Also leads back to an indeterminate term, so you can it again the rule of L'Hospital to apply, possibly resulting in a finite number of steps to the goal.

The reversal of the rule does not apply: From the fact that the limit exists, does not follow necessarily that also exists.

Precise formulation

Be a non-empty open interval and be differentiable functions ( goes from below ) for both converge to 0 or both diverge determined.

If it is true for all, and converges to a value determined or diverges, so does. The same applies if you everywhere ( going from top to ) replaced.

If proper subset of an open interval on which the above requirements are met, ie especially

The rate also applies to improper interval limits.

Sketch of proof

In the case, the functions and continue in place by constantly. The sentence can therefore be attributed to the extended mean value theorem, according to which there is, under the given conditions for each one, so that

With the border crossing, the assertion follows.

By transformation of variables can be extended to the set to the improper case.

Intuitive Explanation

The rule is based on that can be functions near a point x0 approximated by their tangents.

If so are the tangent equations. Your quotient is therefore an approximation for.

Application Examples

Border crossing at x0 = 0

To examine the convergence or divergence of. Effected by exposing and. It is

If for converges or diverges determined, the rule of L'Hospital may be applied. Now applies

Thus, the rule of L'Hospital is applicable. This follows the convergence of with limit 0

Border crossing at infinity

To examine the convergence or divergence of. It sets and. Both and are definitely divergent.

If for converges or diverges determined, the rule of L'Hospital should be applied. Now applies

That is, is definitely divergent. Therefore, the rule of L'Hospital may be applied. From it the certain divergence follows

Warning examples

Compliance with the conditions

Be and. For there is a case.

However, the rule of L'Hospital can not be applied, because for indefinitely divergent, as a periodic function exists. Despite the failure of the rule of L'Hospital converges. Indeed, it is.

Landau - calculus

If you want to calculate the threshold value and the Taylor expansion of the denominator and numerator knows about, it is often easier to determine the limit on the - calculus, as repeatedly apply the rule of L' Hospital.

So applies, for example.