BackL'Hospital's Rule and Indeterminate Forms
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L'Hospital's Rule
Introduction to L'Hospital's Rule
L'Hospital's Rule is a powerful tool in calculus for evaluating limits that result in indeterminate forms, such as 0/0 or ∞/∞. This rule allows us to differentiate the numerator and denominator separately to find the limit.
Indeterminate Forms: Expressions like 0/0 or ∞/∞ that do not have a clear value and require further analysis.
Applicability: L'Hospital's Rule can only be used when the original limit yields an indeterminate form.
Statement of L'Hospital's Rule
If f(x) and g(x) are differentiable near x = a (except possibly at a), and if:
and , or and
and near x = a
Then,
provided the limit on the right exists or is infinite.
Derivation Using Taylor Expansion (Additional info)
For functions differentiable at x = a:
So,
If and , then:
Examples of L'Hospital's Rule
Example 1:
Direct substitution gives (indeterminate form).
Apply L'Hospital's Rule:
Example 2:
Direct substitution gives (not indeterminate).
No need for L'Hospital's Rule.
Example 3:
As , and , so the product is indeterminate of the form .
Rewrite as a ratio:
Apply L'Hospital's Rule:
Summary Table: Indeterminate Forms and L'Hospital's Rule
Form | Indeterminate? | Can Use L'Hospital's Rule? |
|---|---|---|
0/0 | Yes | Yes |
∞/∞ | Yes | Yes |
1/∞ | No | No |
0 | No | No |
Additional info: L'Hospital's Rule can also be applied repeatedly if the first application still yields an indeterminate form. Always check the form before applying the rule.
