Flashcards in 12.2.3 L'Hôpital's Rule and One to the Infinite Power Deck (7):
L'Hôpital's Rule and One to the Infinite Power
• Some indeterminate forms have to be transformed before you can apply L’Hôpital’s rule.
• In order to apply L’Hôpital’s rule to a limit of the form , use the properties of logarithms to rewrite the exponent as a logarithm.
- You may encounter a limit that produces one to the infinite power, , which is another indeterminate form. It could be one, because one to any power is one. Or it could be infinity, because it began as one and a tiny bit more, which grows large when raised to infinity.
- If you encounter a limit that produces this form, you will need to transform the expression. The key is to raise the number e to the natural log of the expression. This equals the original expression.
- Once you have transformed the original limit, you can focus on the expression to which e is raised.
- This new limit does not equal the original limit. It is a
sub-problem. It produces an indeterminate product, which you must transform into an indeterminate quotient.
- Now you can use L’Hôpital’s rule. The sub-problem limit equals one.
- By plugging in the value of the limit into the sub-problem, you can evaluate the original limit. Since e raised to the first power is still e, that’s your answer.
- Some mathematicians use this limit expression as an alternate definition for e.
Evaluate limx →0+ (1+3x)^1/2x
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