Limit x→0+ x^(1/x - 1): Is It Defined?

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In summary, the conversation discusses a limit problem where the limit is calculated to be 0, but a calculator gave a different result. The person asks if they are correct, and another person confirms their solution.
  • #1
daniel_i_l
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Homework Statement


I was doing a problem and got to the following limit:
[tex]lim_{x \rightarrow 0^{+}} x^{\frac{1}{x} - 1}[/tex] I calculated it and got 0 but when I calculated it here:
http://wims.unice.fr/wims/en_tool~analysis~function.en.html
It said that it wans't defined. Am I right?
Thanks.

Homework Equations


The Attempt at a Solution

 
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  • #2
try the following:
[tex]lim_{x \rightarrow 0^{+}} x^{\frac{1}{x} - 1}=\lim_{x\rightarrow 0^{+}}e^{ln(x)^{\frac{1}{x}-1}}=\lim_{x\rightarrow\ 0^{+}} e^{\frac{1-x}{x}ln(x)}=e^{\lim_{x\rightarrow\ 0^{+}}\frac{1-x}{x} \lim_{x\rightarrow\ 0^{+}}ln(x)}=e^{\infty*(-\infty)}=e^{-\infty}=0[/tex]
 
  • #3
Thanks, that's what I did but the wims calculator confused me.
 

1. What is the limit of x^(1/x - 1) as x approaches 0 from the right?

The limit of x^(1/x - 1) as x approaches 0 from the right is undefined. This is because as x approaches 0, the value of x^(1/x - 1) becomes increasingly large and approaches infinity.

2. Can the limit of x^(1/x - 1) as x approaches 0 from the right be calculated using L'Hopital's rule?

No, L'Hopital's rule cannot be used to calculate the limit of x^(1/x - 1) as x approaches 0 from the right. This is because the limit is undefined and L'Hopital's rule can only be applied to functions with finite limits.

3. How can the limit of x^(1/x - 1) as x approaches 0 from the right be visualized?

The graph of x^(1/x - 1) as x approaches 0 from the right has a vertical asymptote at x=0. This means that the function approaches infinity as x approaches 0 from the right.

4. Is the limit of x^(1/x - 1) as x approaches 0 from the right equal to 0?

No, the limit of x^(1/x - 1) as x approaches 0 from the right is not equal to 0. As mentioned before, the limit is undefined and the function approaches infinity as x approaches 0 from the right.

5. Can the limit of x^(1/x - 1) as x approaches 0 from the right be evaluated using a calculator?

No, most calculators are not able to handle limits of functions that approach infinity. In order to evaluate this limit, we must use mathematical techniques such as L'Hopital's rule or graphing to determine the behavior of the function as x approaches 0 from the right.

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