Flashcards in 12.2.4 Another Example of One to the Infinite Power Deck (10):
Another Example of 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.
- When you encounter the indeterminate form , you will need to make use of two facts about exponents and logarithms.
- The first is that e raised to the natural log of any expression is equal to that same expression.
- The second is that when there is an exponent inside a natural log expression, it can be moved to the outside as a factor.
- Now that you have rewritten the expression, you can evaluate an easier limit. Forget about e and take the limit of its exponent.
- Remember that this sub-problem is not equal to the original limit. It is just a side calculation.
- To evaluate the limit in the sub-problem, you will have to transform the expression to produce an indeterminate
quotient. Then you can apply L’Hôpital’s rule.
- The limit from the sub-problem is equal to –1, but that is not the value of the original limit!
- When you plug in the result of the side calculation, you get the value of the original limit.
Evaluate limx→0+ x^tanx
Evaluate limx→0+ (cotx)^sinx
Evaluate limx→0 (1–x)^1/5x.
Evaluate limx→2 (x^2)^1/ln(x–1).
Evaluate limx→0+ x^1/1+lnx.
Evaluate limx→1+ (x – 1)^lnx.
Evaluate limx→∞ (lnx)^1/x.