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Calc I: Raising Limits to Functional Exponents

  • #1

Homework Statement


Suppose ##f(x)## and ##g(x)## [itex]\rightarrow[/itex] 0 as x [itex]\rightarrow[/itex] 0+. Find examples of functions f and g with these properties and such that:

a.) ## \lim_{x\rightarrow 0+} { f(x)^{g(x)} = 0 } ##


Homework Equations


None

The Attempt at a Solution



Let ## f(x) = 2^x-1 ## and ## g(x) = x ##

## \displaystyle \lim_{x\rightarrow 0+} ({2^x-1})^{x} = (2^{x^2}-1^{x}) = (1-1) = 0##

My experience with limits is basic, recently while experimenting with expressions approaching infinity I was told that if you obtain an indeterminate then you need to further simplify the expression, in this case substituting x for zero immediately reduced my expression to one which is an determinate. I imagine this means there is no need to simplify and that my solution is invalid, therefore I am confused on how to proceed.

Thank you in advanced for the help.
 

Answers and Replies

  • #2
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## \displaystyle \lim_{x\rightarrow 0+} ({2^x-1})^{x} = (2^{x^2}-1^{x}) = (1-1) = 0##
Take another look at this. What did you do wrong? :tongue:
 
  • #3
Ray Vickson
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Homework Statement


Suppose ##f(x)## and ##g(x)## [itex]\rightarrow[/itex] 0 as x [itex]\rightarrow[/itex] 0+. Find examples of functions f and g with these properties and such that:

a.) ## \lim_{x\rightarrow 0+} { f(x)^{g(x)} = 0 } ##


Homework Equations


None

The Attempt at a Solution



Let ## f(x) = x^2-1 ## and ## g(x) = x ##

## \displaystyle \lim_{x\rightarrow 0+} ({2^x-1})^{x} = (2^{x^2}-1^{x}) = (1-1) = 0##

My experience with limits is basic, recently while experimenting with expressions approaching infinity I was told that if you obtain an indeterminate then you need to further simplify the expression, in this case substituting x for zero immediately reduced my expression to one which is an determinate. I imagine this means there is no need to simplify and that my solution is invalid, therefore I am confused on how to proceed.

Thank you in advanced for the help.
Your "equation" ##(2^x - 1)^x = 2^{x^2} - 1^x## is false. For example, when x = 2 the left-hand-side is 9 while the right-hand-side is 15.

In general we have
[tex] (2^x-1)^x = \sum_{n=0}^{\infty} {x \choose n} (-1)^n \, 2^{(2-n)x}.[/tex] The series is finite if x is a positive integer, and is infinite if x > 0 is non-integer.
 

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