Home > L'Hôpital's rule

(L'Hôpital is commonly spelled as both "L'Hospital" and "L'Hôpital" since hôpital (French) is equivalent to hospital (English).)

1 Overview

When determining the limit of a quotient f(x)/g(x) when both the numerator and denominator approach 0 or infinity, l'Hôpital's rule states that differentiation of both the numerator and denominator does not change the limit. This differentiation, however, often simplifies the quotient and/or converts it to a determinate form, allowing the limit to be determined more easily.

Symbolically, if

  or  

then

2 Examples

• Here is an example involving 0/0:

However, it is simpler to observe that this limit is just the definition of the derivative of sin(x) at x = 0.

In fact this particular limit is needed in the most usual proof that the derivative of sin(x) is cos(x), but we cannot use l'Hôpital's rule to do this, as it would produce a circular argument.

• Here is a more elaborate example involving the indeterminate form 0/0. Appplying the rule a single time still results in an indeterminate form. In this case, the limit may be evaluated by applying L'Hopital's rule three times:

• Here is another case involving 0/0.

• Here is a case of ∞/∞:

• This one involves ∞/∞. Assume n is a positive integer.

Iterate the above until the exponent is 0. Then one sees that the limit is 0.

• This one involves ∞/∞.

3 Proof

The most common proof of L'Hôpital's rule uses Cauchy's mean value theorem.

According to Cauchy's mean value theorem there is a constant in the interval such that:

Since ,

If , then

Therefore

There are more intuitive proofs of the rule. If

tends to the indeterminate form 0/0, then the rule can be proven with a local linearity argument. If it tends to the indeterminate form , then this can be converted to 0/0 form using the identity : By assuming this limit equals L, and taking the derivative of the numerator and denominator, it can be proven that .

4 Other applications

Many other indeterminate forms, such as , , and can be calculated using l'Hôpital's rule.

For example, to handle a case of , the difference of two functions is converted to a quotient:

5 Other methods of computing limits

Although L'Hôpital's rule's rule is a powerful way of computing otherwise hard to compute limits, it is not always the easiest. Some limits are actually easier to compute using the Taylor series expansion.

For example,