What L'Hopital's Rule Is For
Some limits cannot be evaluated simply by substituting the limiting value into the function. One of the most important examples occurs when substitution produces .
The expression is called an indeterminate form. It does not mean that the limit is zero, and it does not mean that the limit is undefined. Instead, it means that direct substitution has not given us enough information to determine the limit.
L'Hopital's Rule gives us a way to transform a limit into another limit that may be much easier to evaluate.
First: Recognize the 0/0 Form
Before using L'Hopital's Rule, substitute the limiting value directly into the numerator and denominator.
If the numerator approaches 0 and the denominator approaches 0, you have the indeterminate form, and L'Hopital's Rule may be appropriate.
An expression of the form is not automatically equal to zero. The numerator and denominator are both approaching zero, so their relative rates of approach determine the limit. That's exactly what L'Hopital's Rule helps us investigate.
L'Hopital's Rule
If and , and the quotient produces the indeterminate form , then under the conditions of the theorem:
The key idea is simple: differentiate the numerator and denominator separately, then take the limit again.
Notice what you do not do: you do not use the quotient rule. The numerator gets differentiated on its own, and the denominator gets differentiated on its own.
A Step-by-Step Method
- Substitute the limiting value. Check what happens to the numerator and denominator.
- Confirm the form is . Do not apply L'Hopital's Rule just because a limit looks difficult.
- Differentiate the numerator and denominator separately.
- Evaluate the new limit. Sometimes the new limit can be evaluated immediately.
- If you still get , apply L'Hopital's Rule again. Multiple applications can be necessary.
Worked Examples
Evaluate .
Evaluate .
Evaluate .
Evaluate .
Evaluate .
When You Should Not Use L'Hopital's Rule
L'Hopital's Rule is specifically designed for quotient limits that produce certain indeterminate forms. In this tutorial, the focus is the form.
If direct substitution produces an ordinary number such as , there is no reason to use L'Hopital's Rule. The limit has already been found.
The difficulty of a limit is not what determines whether L'Hopital's Rule applies. The form of the limit after substitution does. Always check the limit first.
L'Hopital's Rule says to differentiate the numerator and denominator separately. For , you get —not the quotient-rule expression.
After differentiating, you still have a limit to evaluate. If the new expression gives another form, you may need to apply the rule again.
Why L'Hopital's Rule Works
At a conceptual level, L'Hopital's Rule compares the rates at which the numerator and denominator approach zero.
Suppose both functions approach zero as . The important question is not simply "do both go to zero?" It is "which one goes to zero faster?" Differentiating the functions reveals information about those rates of change.
This is why L'Hopital's Rule can turn a difficult quotient into a much simpler one. Instead of comparing the original functions directly, we compare their derivatives.
For AP Calculus purposes, you usually do not need to prove the theorem. You do need to recognize when its conditions are met and apply it correctly.
Practice Problems
Evaluate .
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Evaluate .
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Evaluate .
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Evaluate .
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