L'Hopital's Rule Is a Quotient Rule
L'Hopital's Rule is designed specifically for limits that produce the indeterminate quotient forms or . When a limit has one of these forms, you can differentiate the numerator and denominator separately and try the limit again.
But several other expressions can also produce indeterminate forms. These include products, differences, and powers. L'Hopital's Rule still helps with them, but you first have to rewrite the expression as a quotient or use logarithms.
The Five Other Indeterminate Forms
The main forms you should recognize are:
The word indeterminate is important. These forms do not tell you what the limit equals. For example, can approach many different values depending on the functions involved.
The goal is therefore not to memorize a numerical answer for each form. Instead, recognize the form and know how to transform it.
1. The Form
A product can produce the indeterminate form . L'Hopital's Rule cannot be applied directly to a product, so rewrite one factor as a reciprocal.
Once the expression has been rewritten as a quotient, check whether the new numerator and denominator produce or . If they do, L'Hopital's Rule is available.
Evaluate .
2. The Form
A difference can produce the indeterminate form . This happens when both terms become infinitely large, but their difference may approach a finite number.
The most common strategy is to combine the terms into a single fraction. Algebraic manipulation, a common denominator, or rationalization can reveal a quotient of the form or .
Evaluate .
3. The Form
Expressions involving powers require a different strategy. If a limit has the form , taking the logarithm converts the exponent into a product.
The key idea is to introduce a new variable:
Then take the natural logarithm:
The original limit can be recovered by exponentiating the final result.
Evaluate .
4. The Form
The form looks different from , but the same logarithm technique works.
If the base approaches zero and the exponent approaches zero, set the entire expression equal to a new variable and take the natural logarithm.
Evaluate .
5. The Form
The form is handled exactly the same way. The exponent approaches zero, so taking logarithms turns the problem into a product or quotient that can be manipulated into a form suitable for L'Hopital's Rule.
Evaluate .
The Big Picture: Transform First, Differentiate Second
The most important skill here is recognizing that L'Hopital's Rule does not apply directly to every indeterminate form.
Instead, your job is to transform the expression into a form where the rule is legal.
For the power forms, logarithms are usually the cleanest route:
A Reliable AP Calculus Strategy
- Substitute first. Determine what form the limit actually has.
- Identify the indeterminate form. Do not automatically assume L'Hopital's Rule applies.
- If you have : rewrite one factor as a reciprocal to create a quotient.
- If you have : combine the terms, rationalize, or otherwise turn the difference into a quotient.
- If you have , , or : take logarithms first.
- Apply L'Hopital's Rule only after obtaining or .
- Check whether another application is necessary. Sometimes the new limit is still indeterminate.
- For logarithmic transformations, exponentiate at the end.
Common Mistakes
L'Hopital's Rule is not a product rule. You cannot differentiate the two factors separately just because the limit gives . Rewrite the product as a quotient first.
The expression is indeterminate. Two quantities can both grow without bound while their difference approaches a finite number, infinity, negative infinity, or something else. You must transform the expression before evaluating it.
When you define and calculate , you have found the logarithm of the desired limit—not the original limit itself. If , then the original limit is .
For real-valued logarithms, the base of the original power expression must be positive in the relevant neighborhood. In standard AP Calculus problems involving expressions such as , this condition is typically satisfied near the limit.
The new quotient can still be or . If it is, another application may be necessary. Always evaluate the new limit before deciding you are finished.
Practice Problems
Evaluate .
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Evaluate .
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Evaluate .
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Evaluate .
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Evaluate .
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