L'Hôpital's Rule
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Alternate Names For L'Hôpital's Rule |
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L'Hospital's Rule |
Bernoulli's Rule |
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L'Hôpital's Rule |
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If the limit \(\displaystyle{ \lim_{x \to c}{\frac{n(x)}{d(x)}} }\) is indeterminate of the type \(0/0\) then |
Quick Notes
This rule can be repeated until a determinate form is found. |
\(x\) can approach a finite value \(c\), \(\infty\) or \(-\infty\) |
For details about determinate and indeterminate forms, click here. |
This is not the quotient rule. For information about the difference, click here. |
Do not use this on determinate forms. It will often give you the wrong answer. |
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L'Hôpital's Rule is used to evaluate a limit when other techniques will not work, like factoring and rationalizing. It especially convenient to use with exponentials and, sometimes, limits involving trig functions. In fact, L'Hôpital's Rule can usually be used on limits where those other techniques work and it is often easier than factoring. So it is a nice tool to have when evaluating limits.
Some Things To Notice |
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1. Be careful to use L'Hôpital's Rule only on limits of indeterminate form. If you use it on a determinate form, you may ( and probably will ) get an incorrect answer.
2. L'Hôpital's Rule can be used multiple times. So, if it doesn't work the first time, check that you still have an indeterminate form, and then use it again.
3. One of the big mistakes that students make when they are first learning L'Hôpital's Rule is to confuse it with the derivative of the function itself. Read this next section to clarify it in your mind.
Difference Between L'Hôpital's Rule and the Quotient Rule
Difference Between L'Hôpital's Rule and the Quotient Rule
It is very easy to confuse L'Hôpital's Rule with the
quotient rule since they are so similar. Here is a rundown of the two for comparison.
1. Quotient Rule - The quotient rule is used on a quotient or ratio of terms to calculate the derivative of a function. The result of the quotient rule is the slope of the original function at all points along the curve. So you can use the result to determine the slope, calculate the equation of a tangent line, find extrema and continue on to determine the second derivative, to name a few uses.
To calculate the quotient rule, we use the following equations. Given that we have a function in the form \(\displaystyle{ f(x) = \frac{n(x)}{d(x)}}\), the derivative \(f(x)\) using the quotient rule is
\(\displaystyle{ f'(x) = \frac{d(x)n'(x) - n(x)d'(x)}{[d(x)]^2}}\)
2. L'Hôpital's Rule - L'Hôpital's Rule is used only when we want to calculate a limit and can be used only under very specific circumstances (see the discussion below for a complete explanation). L'Hôpital's Rule works like this.
If we have a limit that goes to an indeterminate form, for example \(\displaystyle{ \lim_{x \to c}{\left[ \frac{n(x)}{d(x)} \right]}}\) where \(\displaystyle{ \lim_{x \to c}{[n(x)]} = 0}\) and \(\displaystyle{ \lim_{x \to c}{[d(x)]} = 0}\) giving us \( 0/0 \) in the original limit, then L'Hôpital's Rule tells us that \(\displaystyle{ \lim_{x \to c}{\frac{n(x)}{d(x)}} = \lim_{x \to c}{\frac{n'(x)}{d'(x)}}}\).
Okay, now let's compare the two using \(\displaystyle{ f(x) = \frac{n(x)}{d(x)}}\)
Quotient Rule | L'Hôpital's Rule | |
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\(\displaystyle{ f'(x) = \frac{d(x)n'(x) - n(x)d'(x)}{[d(x)]^2} }\) | \(\displaystyle{ \lim_{x \to c}{[f(x)]} = \lim_{x \to c}{\frac{n(x)}{d(x)}} = \lim_{x \to c}{\frac{n'(x)}{d'(x)}} }\) | |
Context: Derivatives | Context: Limits | |
One of several ways to calculate the derivative of a function. | Can be used only on indeterminate limits. | |
The resulting equation has many uses including calculating the slope of a curve. |
The resulting fraction \( [n'(x)/d'(x)]\) has no meaning other than its limit is the same as the limit of the original function. |
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Study Note - When you are learning two similar concepts or you are learning a new concept that is similar to one you already know and you are getting confused, try the technique we used above (setting them side-by-side and comparing and contrasting them). This will help you separate them in your mind and know when to use which technique. This also works for learning similar words in foreign languages. Click here for more study techniques.
If you haven't already, now would be a good time to study what is meant by an indeterminate form, since the following discussion requires you to know it. You can find the complete discussion on the indeterminate forms page.
Indeterminate Forms Table
Determinate-Indeterminate Forms Table | ||
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Indeterminate Forms | Determinate Forms | |
\( 0/0 \) | \( \infty + \infty = \infty \) | |
\( \pm \infty / \pm \infty \) | \( - \infty - \infty = - \infty\) | |
\( \infty - \infty \) | \( 0^{\infty} = 0 \) | |
\( 0 (\infty) \) | \( 0^{-\infty} = \infty \) | |
\( 0^0 \) | \( (\infty) \cdot (\infty) = \infty \) | |
\( 1^{\infty} \) | ||
\( \infty ^ 0 \) | ||
Use L'Hôpital's Rule | Do Not Use L'Hôpital's Rule | |
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Types of Indeterminate Forms |
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Notice in the L'Hôpital's Rule equation, we require the limit equation to be in a fraction. This is a strict requirement that cannot be broken. If the limit equation is not a fraction, we cannot use L'Hôpital's Rule. This section contains a discussion of the four main types of equations that you will run across where you need to determine if the limit is an indeterminate form. (This list is also found on the indeterminate forms page. )
Case 1 - - Indeterminate Quotients |
Case 2 - - Indeterminate Products |
Case 3 - - Indeterminate Differences |
Case 4 - - Indeterminate Powers |
Okay, now that you know something about L'Hôpital's Rule, under what conditions you can use it and various situations you will encounter, let's watch some videos to help explain how it works and see some examples.