12.2.3 L'Hôpital's Rule and One to the Infinite Power

Question 1

L’Hôpital’s Rule and One to the Infinite Power

Answer 1

• 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.

Question 2

note

Answer 2

• 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.

Question 3

Evaluate limx →0+ (1+3x)^1/2x

Answer 3

e^ 3/2

Question 4

Evaluate limx→0+ x^x

Answer 4

1

Question 5

Evaluate lim x→∞ (1+5/x)^x

Answer 5

e^5

Question 6

Evaluate limx→∞(1+1/x^2)x.

Answer 6

1

Question 7

Evaluate limx→0(1+2x+x^2)1/x

Answer 7

e^ 2