L'Hospital rule: Difference between revisions

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This rule tells us that if we have a quotient <math>f(x)/g(x)</math> and the limit results in an indeterminate form <math>0/0</math> or <math>\infty/\infty</math>, then we can safely assume that the limit of the quotient <math>f/g</math> is equal to the limit of <math>f'/g'</math>. That is, the limit of the quotient of the derivatives. All under the condition that the functions are differentiable. If the limit of <math>f'/g'</math> also results in an indeterminate form, we can apply l'Hospital again.
This rule tells us that if we have a quotient <math>f(x)/g(x)</math> and '''the limit results in an indeterminate form''' <math>0/0</math> or <math>\infty/\infty</math>, then we can safely assume that the limit of the quotient <math>f/g</math> is equal to the limit of <math>f'/g'</math>. That is, '''the limit of the quotient of the derivatives'''. All under the condition that the functions are differentiable. If the limit of <math>f'/g'</math> also results in an indeterminate form, we can apply l'Hospital again.


The graphical idea is that if both functions are differentiable, then near a point they both can be approximated by their respective linear approximations. If both limits are converging to zero, with the functions being differentiable, it should be safe to assume that their respective linear approximations are also converging to zero.
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>\lim_{x \ \to \ a} f(x)/g(x) = n</math>, where <math>n \neq 0</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>n/n = 1</math>. Now with derivatives we can find critical points, maximum or minimum of a function. For other points there is no relationship between a function and its derivative. If at <math>x = a</math> the limit of a function is some number, what guarantees that its derivative have the same limit? Nothing. When does a function's limit is zero? When we are calculating a limit that coincides with the function's root or when it's an asymptote. We learn that when <math>f'(x) = 0</math> that point is either a maximum, minimum or an inflection point. In case the limit goes to infinity, the tangent is going to 90°, which is indeterminate. I hope this makes it clear why l'Hospital applies to a very specific case of limits.


In the same way that a linear approximation may make computations easier by replacing a complicated equation with a simpler one, L'Hospital rule replaces the algebraic manipulations that we'd have to do with calculating a quotient of derivatives. With this rule it becomes trivial to show that <math>\lim_{x \ \to \ 0} \frac{\sin(x)}{x} = 1</math> for example.
One may naturally think that we can extend the rule to multivariable functions. No, it 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>f(x,y)/g(x,y)</math> and an indeterminate form with a limit. We cannot consider <math>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.


'''Careful!''' This rule is exclusive to the specific cases of limits where we have <math>0/0</math> or <math>\infty/\infty</math>. Don't blindly use it thinking that we can use it anywhere. For example, consider <math>\lim_{x \ \to \ 0} \frac{x + 1}{x^2 + 2}</math>. We don't need l'Hospital and if we apply it, the result is going to be wrong.
==Proof of l'Hospital==
 
One may naturally think that we can extend the rule to multivariable functions. We cannot do that because with multivariable functions we have partial derivatives, which represent rates of change on a per variable basis. 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 in 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>f(x,y)/g(x,y)</math> and an indeterminate form with a limit. We cannot consider <math>f'(x,y)/g'(x,y)</math> because there isn't the concept of differentiation in both variables at the same time.

Revision as of 23:25, 3 April 2022

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. 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.

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]. 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 points, maximum or minimum of a function. For other points there is no relationship between a function and its derivative. If at [math]\displaystyle{ x = a }[/math] the limit of a function is some number, what guarantees that its derivative have the same limit? Nothing. When does a function's limit is zero? When we are calculating a limit that coincides with the function's root or when it's an asymptote. We learn that when [math]\displaystyle{ f'(x) = 0 }[/math] that point is either a maximum, minimum or an inflection point. In case the limit goes to infinity, the tangent is going to 90°, which is indeterminate. 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, it 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.

Proof of l'Hospital

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