L'Hospital rule
This rule tells us that if we have a quotient [math]\displaystyle{ f(x)/g(x) }[/math] and the limit results in an indeterminate form [math]\displaystyle{ 0/0 }[/math] or [math]\displaystyle{ \infty/\infty }[/math], then we can safely assume that the limit of the quotient [math]\displaystyle{ f/g }[/math] is equal to the limit of [math]\displaystyle{ f'/g' }[/math]. That is, the limit of the quotient of the derivatives (not the limit of the derivative of the quotient!!!). All under the condition that the functions are differentiable. If the limit of [math]\displaystyle{ f'/g' }[/math] also results in an indeterminate form, we can apply l'Hospital again. [math]\displaystyle{ 0/\infty }[/math] or vice-versa may show up but we cannot apply l'Hospital in this case!
One would naturally think that this rule can be applied to any limit. No, it can't. Think about this: suppose that we have [math]\displaystyle{ \lim_{x \ \to \ a} f(x)/g(x) = n }[/math], where [math]\displaystyle{ n \neq 0 }[/math] and [math]\displaystyle{ n \neq 1 }[/math]. From this we can already conclude one thing: the limits of both functions can't be equal to each other. Otherwise the limit would be [math]\displaystyle{ n/n = 1 }[/math]. Now with derivatives we can find critical, maximum or minimum points of a function. Such points always coincide with [math]\displaystyle{ f'(x) = 0 }[/math]. They can also coincide with [math]\displaystyle{ f(x) = 0 }[/math].
In case both functions go to infinity the natural question is: which one is faster or slower? This naturally suggests analysing their respective derivatives. That's the idea of this rule, to analyse how fast one function is going to the same point in comparison to the other. I hope this makes it clear why l'Hospital applies to a very specific case of limits.
One may naturally think that we can extend the rule to multivariable functions. No, we can't. We don't have an equivalent to the l'Hospítal rule for two or more variables. The reason for this lies in the fact that multiple paths lead to different limits with two or more variables. For multivariable functions there doesn't exist the concept of a derivative that is detached from direction. Suppose that we have a quotient [math]\displaystyle{ f(x,y)/g(x,y) }[/math] and an indeterminate form with a limit. We cannot consider [math]\displaystyle{ f'(x,y)/g'(x,y) }[/math] because while it may happen that the same path leads both functions to an indeterminate form, we don't have a tool to treat all paths at the same time. We have the directional derivative but even then there comes another issue: what do we do if one variable changes faster than the other? Even worse, such question depends on the path you take itself.
Note: this rule does not solve all problems. Certain cases are pathological because we have functions which can be differentiated indefinitely such as sine or exp and replacing the functions by their respective derivatives doesn't simplify anything. Such cases require some other procedure such as substituting variables.
Proof of l'Hospital
Before going on, let's establish conditions for the formula to hold:
- [math]\displaystyle{ g'(x) \neq 0 }[/math] on an open interval containing [math]\displaystyle{ a }[/math]. The limit may be finite and equal to zero, what we can't have is the derivative itself assuming the value zero.
- The indeterminate form must be [math]\displaystyle{ 0/0 }[/math] or [math]\displaystyle{ \infty/\infty }[/math]. Others such as [math]\displaystyle{ 0^0 }[/math] or [math]\displaystyle{ 1^{\infty} }[/math] are not considered.
- [math]\displaystyle{ \lim_{x \ \to \ a} \frac{f'(x)}{g'(x)} }[/math] must exist. If the limit doesn't exist it we can't rely on faith or pure guessing to attempt to find it.
Let's use the Cauchy's Mean Value Theorem and recall that a derivative is, by definition, a limit:
[math]\displaystyle{ \frac{f'(a)}{g'(a)} = \frac{\lim_{x \to a} \frac{f(x) - f(a)}{x - a}}{\lim_{x \to a} \frac{g(x) - g(a)}{x - a}} }[/math]
[math]\displaystyle{ = \lim_{x \to a} \frac{f(x) - f(a)}{g(x) - g(a)} }[/math]
[math]\displaystyle{ = \lim_{x \to a} \frac{f(x) - 0}{g(x) - 0} }[/math] (from the conditions to apply the rule we know that [math]\displaystyle{ f(a) = g(a) = 0 }[/math])
[math]\displaystyle{ = \lim_{x \to a} \frac{f(x)}{g(x)} }[/math]
With this we have proven the formula.
Links for the proofs:
- https://brilliant.org/wiki/lhopitals-rule/
- https://en.wikipedia.org/wiki/L'H%C3%B4pital's_rule
- https://www.khanacademy.org/math/ap-calculus-ab/ab-diff-contextual-applications-new/ab-4-7/a/proof-of-special-case-of-lhpitals-rule
- https://proofwiki.org/wiki/L%27H%C3%B4pital%27s_Rule
- https://artofproblemsolving.com/wiki/index.php/L%27H%C3%B4pital%27s_Rule
