L'Hôpital's rule

In calculus, l'Hôpital's rule (alternatively, l'Hospital's rule) uses derivatives to help compute limits with indeterminate forms. Application (or repeated application) of the rule often converts an indeterminate form to a determinate form, allowing easy computation of the limit. The rule is named after the 17th-century French mathematician Guillaume de l'Hôpital, who published the rule in his book Analyse des infiniment petits pour l'intelligence des lignes courbes (1696), the first book to be written on differential calculus.

The rule is believed to be the work of Johann Bernoulli. L'Hôpital, a nobleman, paid Bernoulli a retainer of 300 pounds per year to keep him updated on developments in calculus and to solve problems he had. Among these problems was that of limits of indeterminate forms. When l'Hôpital published his book, he gave due credit to Bernoulli and, not wishing to take credit for any of the mathematics in the book, he published the work anonymously. Bernoulli, who was known for being extremely jealous, claimed to be the author of the entire work, and until recently, it was believed to be so. Nevertheless, the rule was named for l'Hôpital, who never claimed to have invented it in the first place.

Overview

Introduction

In simple cases, L'Hôpital's rule states that for functions $f(x)$ and $g(x)$:

if $f(c)=g(c)=0\,$ or $f(c)=g(c)=\infty$ then:

$\lim _{x\to c}{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f'(x)}{g'(x)}}$

where the prime (') denotes the derivative.

Among other requirements, for this rule to hold, the limit $\lim _{x\to c}{\frac {f'(x)}{g'(x)}}$ must exist. Other requirements are detailed below, in the formal definition.

Formal statement

When determining the limit of a quotient $f(x)/g(x)\$ when both f and g approach 0, or f and g approach infinity, L'Hôpital's rule states that $f'/g'\$ has the same limit (if the limit exists), provided that g′ is nonzero throughout some interval containing the point in question. This differentiation often simplifies the quotient and/or converts it to a determinate form, allowing the limit to be determined more easily.

Symbolically let $\mathbb {R} ^{*}=\mathbb {R} \cup \{\pm \infty \}$. Suppose that $c\in \mathbb {R} ^{*}$, that

$\lim _{x\to c}{f'(x) \over g'(x)}=A,A\in \mathbb {R} ^{*}$

and that $g'(x)\neq 0$ for all $x\neq c$ in an open interval (a,b) containing c (or with $b=\infty$ if $c=\infty$ or with $a=-\infty$ if $c=-\infty$). If

${\begin{cases}\lim _{x\to c}{f(x)}=\lim _{x\to c}g(x)=0\\{\mbox{or}}\\\lim _{x\to c}{|g(x)|}=+\infty \end{cases}}$

then

$\lim _{x\to c}{f(x) \over g(x)}=A.$

L'Hôpital's rule also holds for one-sided limits.

Basic indeterminate forms (all others reduce to these):

${0 \over 0}\qquad {\infty \over \infty }$

Other indeterminate forms:

${\infty \qquad 0\cdot \infty \qquad 0^{0}\qquad \infty -\infty \qquad }$

Note the requirement that the limit $\lim _{x\to c}{\frac {f'(x)}{g'(x)}}$ exists. Differentiation of limits of this form can sometimes lead to limits that do not exist. In that case, L'Hôpital's rule cannot be applied. For instance if $f(x)=x+\sin(x)$ and $g(x)=x$, then

$\lim _{x\to \infty }{\frac {f'(x)}{g'(x)}}=\lim _{x\to \infty }(1+\cos(x))$

does not exist, whereas

$\lim _{x\to \infty }{\frac {f(x)}{g(x)}}=1.$

In practice one often uses the rule and, if the resulting limit exists, concludes that it was legitimate to use L'Hôpital's rule.

Note also the requirement that the derivative of g not vanish throughout an entire interval containing the point c. Without such a hypothesis, the conclusion is false. Thus one must not use L'Hôpital's rule if the denominator oscillates wildly near the point where one is trying to find the limit. For example if $f(x)=x+\cos(x)\sin(x)$ and $g(x)=e^{\sin(x)}(x+\cos(x)\sin(x))$, then

$\lim _{x\to \infty }{\frac {f'(x)}{g'(x)}}$	$=\lim _{x\to \infty }{\frac {2\cos ^{2}{x}}{e^{\sin(x)}\cos(x)(x+\sin(x)\cos(x)+2\cos(x))}}$
	$=\lim _{x\to \infty }{\frac {2\cos(x)}{e^{\sin(x)}(x+\sin(x)\cos(x)+2\cos(x))}}=0$

whereas

$\lim _{x\to \infty }{\frac {f(x)}{g(x)}}=\lim _{x\to \infty }{\frac {1}{e^{\sin(x)}}}$

does not exist since ${\frac {1}{e^{\sin(x)}}}$ fluctuates between e^-1 and e.

Examples

• Here is an example involving the sinc function, which has the form 0/0 :

$\lim _{x\to 0}\mathrm {sinc} (x)\,$	$=\lim _{x\to 0}{\frac {\sin \pi x}{\pi x}}\,$	$=\lim _{x\to 0}{\frac {\sin x}{x}}\,$
		$=\lim _{x\to 0}{\frac {\cos x}{1}}={\frac {1}{1}}=1\,$

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. Applying the rule a single time still results in an indeterminate form. In this case, the limit may be evaluated by applying l'Hôpital's rule three times:

$\lim _{x\to 0}{2\sin x-\sin 2x \over x-\sin x}$	$=\lim _{x\to 0}{2\cos x-2\cos 2x \over 1-\cos x}$
	$=\lim _{x\to 0}{-2\sin x+4\sin 2x \over \sin x}$
	$=\lim _{x\to 0}{-2\cos x+8\cos 2x \over \cos x}$
	$={-2\cos 0+8\cos 0 \over \cos 0}$
	$=6\,$

• Here is another case involving 0/0:

$\lim _{x\to 0}{e^{x}-1-x \over x^{2}}=\lim _{x\to 0}{e^{x}-1 \over 2x}=\lim _{x\to 0}{e^{x} \over 2}={1 \over 2}$

• Here is a case of ∞/∞:

$\lim _{x\to \infty }{\frac {\sqrt {x}}{\ln(x)}}=\lim _{x\to \infty }{\frac {\ 1/(2{\sqrt {x}}\,)\ }{1/x}}=\lim _{x\to \infty }{\frac {\sqrt {x}}{4}}=\infty$

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

$\lim _{x\to \infty }x^{n}e^{-x}=\lim _{x\to \infty }{x^{n} \over e^{x}}=\lim _{x\to \infty }{nx^{n-1} \over e^{x}}=n\lim _{x\to \infty }{x^{n-1} \over e^{x}}$

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

• This one also involves ∞/∞:

$\lim _{x\to 0+}x\ln x=\lim _{x\to 0+}{\ln x \over 1/x}=\lim _{x\to 0+}{1/x \over -1/x^{2}}=\lim _{x\to 0+}-x=0$

• This is the impulse response of a raised cosine filter:

$\lim _{t\to 0}\,\mathrm {sinc} (f_{0}t)\cdot {\frac {\cos \left(\pi \alpha f_{0}t\right)}{\left[1-\left(2\alpha f_{0}t\right)^{2}\right]}}$	$=\left\{\lim _{t\to 0}\,\mathrm {sinc} (f_{0}t)\right\}\cdot \left.{\frac {\cos \left(\pi \alpha f_{0}t\right)}{\left[1-\left(2\alpha f_{0}t\right)^{2}\right]}}\,\right|_{t=0}$
	$=1\cdot 1=1$

• And:

$\lim _{t\to {\frac {1}{2\alpha f_{0}}}}\mathrm {sinc} (f_{0}t)\cdot {\frac {\cos \left(\pi \alpha f_{0}t\right)}{\left[1-\left(2\alpha f_{0}t\right)^{2}\right]}}$	$=\mathrm {sinc} \left({\frac {1}{2\alpha }}\right)\cdot \lim _{t\to {\frac {1}{2\alpha f_{0}}}}{\frac {\cos \left(\pi \alpha f_{0}t\right)}{\left[1-\left(2\alpha f_{0}t\right)^{2}\right]}}$
	$=\mathrm {sinc} \left({\frac {1}{2\alpha }}\right)\cdot \left({\frac {-\pi /2}{-2}}\right)$
	$=\sin \left({\frac {\pi }{2\alpha }}\right)\cdot {\frac {\alpha }{2}}$

General proof by definition of derivative

$\lim _{x\to c}{f(x) \over g(x)}=\lim _{e\to 0}{f(c+e) \over g(c+e)}$

Or, in other words, when e is infinitely small, then

${f(c+e) \over g(c+e)}={f(c) \over g(c)},$

using a/b = c/d = (a − c)/(b − d) we can write, for e approaching 0,

${f(x) \over g(x)}={[f(c+e)-f(c)] \over [g(c+e)-g(c)]}.$

Dividing both numerator and denominator by e

${f(x) \over g(x)}~={[f(c+e)-f(c)]/e \over [g(c+e)-g(c)]/e},$

taking the limit when e approaches 0 we observe that the numerator and denominator on the right hand side of the equation are actually f′(c) and g′(c) respectively, therefore

$\lim _{x\to c}{f(x) \over g(x)}={f'(c) \over g'(c)}.$

The proof is independent of whether $\lim _{x\to c}f(x)=\lim _{x\to c}g(x)=0$ or $\lim _{x\to c}|g(x)|=\infty$

Proof by Cauchy's mean value theorem

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

1) The case when $f(x)\to 0,g(x)\to 0$

First, we expand continuously (or redefine) $f(x)$ and $g(x)$ by $0$ for $x=c$. This doesn't change the limit since the limit doesn't depend on the value in the point (by definition).

According to Cauchy's mean value theorem there is a constant $\xi$ in $c<\xi <c+h$ such that:

${f'(\xi ) \over g'(\xi )}={f(c+h)-f(c) \over g(c+h)-g(c)}$

Since $f(c)=g(c)=0$,

${f'(\xi ) \over g'(\xi )}={f(c+h) \over g(c+h)}$

If $h\to 0$ then $\xi \to c$ and

$\lim _{x\to c}{f'(x) \over g'(x)}=\lim _{h\to 0}{f'(\xi ) \over g'(\xi )}=\lim _{h\to 0}{f(c+h) \over g(c+h)}=\lim _{x\to c}{f(x) \over g(x)}$

2) The case when $|g(x)|\to +\infty$

Let $x<y<x+h$. Then using Cauchy's mean value theorem:

${f'(\xi ) \over g'(\xi )}={f(x)-f(y) \over g(x)-g(y)}$

We rewrite that in the form

${f(x) \over g(x)}={f(y) \over g(x)}+\left[1-{g(y) \over g(x)}\right]{f'(\xi ) \over g'(\xi )}$

and then by the discussion of the two cases

${\begin{cases}\lim _{x\to c}{f'(x) \over g'(x)}=B\in \mathbb {R} \\\\\lim _{x\to c}{f'(x) \over g'(x)}=\pm \infty \end{cases}}$

we show that the limit of f(x)/g(x) tends to the same when $x\to c$ and $h\to 0$.

Proof by local linearity

Suppose that functions $f(x)$ and $g(x)$ are continuous and differentiable on the interval $(a,b)$. In addition, there is a function $h(x)$ that is equal to the ratio of $f(x)$ and $g(x)$. In other words,

$h(x)={{f(x)} \over {g(x)}}$

When these functions are equal to a real number $c$, which is on the interval $(a,b)$, their ratio

$h(c)={{f(c)} \over {g(c)}}$

For example, if $f(x)=x^{3}+1$ and $g(x)=\sin \left({\frac {\pi x}{4}}\right)$, then $h(x)$ evaluated at 1 is

$h(1)={{f(1)} \over {g(1)}}={{1^{3}+1} \over {\sin \left({\frac {\pi \cdot 1}{4}}\right)}}={{2} \over {1/{\sqrt {2}}}}=2{\sqrt {2}}$

However, suppose that $f(c)=g(c)=0\,$, where $c$ is again a value on the interval $(a,b)$. Then

$h(c)={{f(c)} \over {g(c)}}={0 \over 0}$

0 divided by 0 is indeterminate. This means that $h(c)$ could be any real number or infinity, based on the context. However, $f(x)$ and $g(x)$ are continuous and differentiable at $x=c$, they can be approximate by lines (because of local linearity.

The linear approximation (or linearization) of $f(x)$ and $g(x)$ around $x=c$ is, respectively, $y=f(c)+f'(c)\,(x-c)$ and $y=g(c)+g'(c)\,(x-c)$. The function $h(x)$ near $x=c$, then, can be approximated by

${\lim _{x\rightarrow c}}\ h(c)={\lim _{x\rightarrow c}{{{\mbox{Linearization of }}f\left(x\right)} \over {{\mbox{Linearization of }}g\left(x\right)}}}$

Using the mathematical expression for the functions' linearizations,

${\lim _{x\rightarrow c}}\ h(c)={\lim _{x\rightarrow c}{{f\left(c\right)+f'\left(c\right)\,\left(x-c\right)} \over {g\left(c\right)+g'\left(c\right)\,\left(x-c\right)}}}$

Since $f(c)$ and $g(c)$ and 0 regardless of the value of $x$,

${\lim _{x\rightarrow c}}\ h(c)={\lim _{x\rightarrow c}{{f'\left(c\right)\,\left(x-c\right)} \over {g'\left(c\right)\,\left(x-c\right)}}}$

There is factor of $(x-c)$ in both the numerator and denominator, which can be canceled, revealing

${\lim _{x\rightarrow c}}\ h(c)={\lim _{x\rightarrow c}{{f'\left(c\right)} \over {g'\left(c\right)}}}$

If $f'(c)$ and $g'(c)$ are both real and $g'(c)$ is not zero, then

${\lim _{x\rightarrow c}}\ h(c)={\lim _{x\rightarrow c}}\ {{f(c)} \over {g(c)}}={{f'(c)} \over {g'(c)}}$

If $g'(c)$ approaches 0 but $f'(c)$ does not, then then $h(c)$ does not exist; or more specifically, it is either $\infty$ or $-\infty$ or approaches each from different sides.

If a 0 over 0 form occurs again, L'Hôpital's rule can be used again (unless $f(x)$ and $g(x)$ resemble the zero function $y=0$ around $c$).

This concludes the proof.

Proof that L'Hôpital's can be applied to infinity over infinity

The proof by definition of the derivative, the proof by Cauchy's mean value theorem, and the proof by local linearity all prove that

${\lim _{x\rightarrow c}}\ {{f(c)} \over {g(c)}}={\lim _{x\rightarrow c}}\ {{f'(c)} \over {g'(c)}}$

when $f(c)=g(c)=0$. Suppose that, instead of approaching zero, $f(c)=g(c)=\pm \infty$. In other words, for any $f(x)$ or $g(x)$ close to $x=c$, $f(x)$ or $g(x)$ gets either larger or smaller as you get closer to $x=c$, depending on whether the function goes to $\infty$ or $-\infty$ respectively. Only the proof by definition of derivative makes provisions for this circumstance. However, it is possible to use the proof for the 0 over 0 form have L'Hôpital's rule apply to $\infty$ over $\infty$.

The $\infty$ over $\infty$ form (which is indeterminate: it can mean any value) can be treated similarly when $f(c)=g(c)=\pm \infty$.

${\lim _{x\rightarrow c}}\ {\frac {f(x)}{g(x)}}={\lim _{x\rightarrow c}}\ {\frac {1/g(x)}{1/f(x)}}$

transforms an $\infty$ over $\infty$ form into a 0 over 0 form, and the latter form can be evaluated with L'Hôpital's rule. According to the rule (taking the derivative of the numerator and denominator,

${\lim _{x\rightarrow c}}\ {\frac {1/g(x)}{1/f(x)}}={\lim _{x\rightarrow c}}\ {\frac {-g'(x)/\left[g(x)\right]^{2}}{-f'(x)/\left[f(x)\right]^{2}}}$

Simplifying the complex fractions,

${\lim _{x\rightarrow c}}\ {\frac {-g'(x)/\left[g(x)\right]^{2}}{-f'(x)/\left[f(x)\right]^{2}}}={\lim _{x\rightarrow c}}\ {\frac {g'(c)\left[f(x)\right]^{2}}{f'(c)\left[g(x)\right]^{2}}}$

Now, using the property of transitivity and the above statements,

${\lim _{x\rightarrow c}}\ {\frac {f(x)}{g(x)}}={\lim _{x\rightarrow c}}\ {\frac {g'(c)\left[f(x)\right]^{2}}{f'(c)\left[g(x)\right]^{2}}}$

Algebraically manipulating both sides, we find that

${\lim _{x\rightarrow c}}\ {\frac {f(x)}{g(x)}}\cdot {\frac {\left[g(x)\right]^{2}}{\left[f(x)\right]^{2}}}={\lim _{x\rightarrow c}}\ {\frac {g'(c)\left[f(x)\right]^{2}}{f'(c)\left[g(x)\right]^{2}}}\cdot {\frac {\left[g(x)\right]^{2}}{\left[f(x)\right]^{2}}}$

${\lim _{x\rightarrow c}}\ {\frac {g(x)}{f(x)}}={\lim _{x\rightarrow c}}\ {\frac {g'(c)}{f'(c)}}$

And finally, when $f(c)=g(c)=\pm \infty$

${\lim _{x\rightarrow c}}\ {\frac {f(x)}{g(x)}}={\lim _{x\rightarrow c}}\ {\frac {f'(c)}{g'(c)}}$

This is L'Hôpital's rule for the $\infty /\infty$. This concludes the proof.

Other applications

Many other indeterminate forms, such as $1^{\infty }$, $\infty ^{0}$, and $\infty -\infty$ can be calculated using l'Hôpital's rule.

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

$\lim _{x\to \infty }x-{\sqrt {x^{2}-x}}=\lim _{x\to \infty }{\frac {\left(x+{\sqrt {x^{2}-x}}\right)\left(x-{\sqrt {x^{2}-x}}\right)}{x+{\sqrt {x^{2}-x}}}}\quad$

$=\lim _{x\to \infty }{\frac {x^{2}-(x^{2}-x)}{x+{\sqrt {x^{2}-x}}}}\quad$

$=\lim _{x\to \infty }{\frac {x}{x+{\sqrt {x^{2}-x}}}}\quad$

$=\lim _{x\to \infty }{\frac {1}{1+{\frac {2x-1}{2{\sqrt {x^{2}-x}}}}}}={\frac {1}{1+1}}={\frac {1}{2}}\quad$

The rule can be used on indeterminate forms involving exponents by using logarithms to "move the exponent down."

Other methods of computing limits

Although l'Hôpital'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,

$\lim _{|x|\to \infty }x\sin {1 \over x}=\lim _{|x|\to \infty }x\left({1 \over x}-{1 \over x^{3}\cdot 3!}+{1 \over x^{5}\cdot 5!}-\cdots \right)\;$

$=\lim _{|x|\to \infty }1-{1 \over x^{2}\cdot 3!}+{1 \over x^{4}\cdot 5!}-\cdots \;=\;1\quad$

Some elementary algebraic manipulation, however, yields:

$\lim _{|x|\to \infty }\ {\sin {1 \over x} \over {1 \over x}}$

And applying l'Hôpital's rule, we have:

$L=\lim _{|x|\to \infty }\ {{\cos {1 \over x}\cdot {-1 \over x^{2}}} \over {-1 \over x^{2}}}$

$=\lim _{|x|\to \infty }{\cos {1 \over x}}\cdot {-1 \over x^{2}}\cdot {x^{2} \over -1}$

$=\cos {1 \over \infty }=\cos {\ 0}=1$

On the other hand, a simple substitution also allows the use of l'Hôpital's rule.

$Let\ u={1 \over x}$ Therefore, as $x\rightarrow \infty$, $u\rightarrow \ 0$

Therefore,

$\lim _{|x|\to \infty }x\sin {1 \over x}\ =\ \lim _{|u|\to \ 0}{1 \over u}\sin {u}\ =\ 1$

Logical circularity

In some cases it may constitute circular reasoning to use l'Hôpital's rule to evaluate such limits as

$\lim _{h\to 0}{(x+h)^{n}-x^{n} \over h}.$

If one uses the evaluation of the limit above for the purpose of proving that

${d \over dx}x^{n}=nx^{n-1}\,$

and one uses l'Hôpital's rule and the fact that

${d \over dx}x^{n}=nx^{n-1}\,$

in the evaluation of the limit, the argument uses the expected proof to prove itself and is therefore fallacious.

External links

• L'Hôpital's rule at Mathworld
• L'Hôpital's rule at planetmath