Lim x->inf t^(1/t)

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  • #1
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Homework Statement



[tex]\lim_{t\rightarrow \infty } t^{\frac{1}{t}}[/tex]

Homework Equations





The Attempt at a Solution



[tex]let \ y = x^{\frac{1}{x}}[/tex]

[tex]\ln y = {\frac{1}{x}} ln x[/tex]

[tex]\lim_{x\rightarrow \infty } ln y = \lim_{x\rightarrow \infty } {\frac{ln x}{x}}[/tex]

L'H
[tex]\lim_{x\rightarrow \infty } ln y = \lim_{x\rightarrow \infty } {\frac{\frac{1}{x}}{1}} = \frac{0}{1} = 0[/tex]

Wolfram shows the answer to be 1 and intuition says that a^0 = 1 so i am not sure where my error is.
 

Answers and Replies

  • #2
Ray Vickson
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Homework Statement



[tex]\lim_{t\rightarrow \infty } t^{\frac{1}{t}}[/tex]

Homework Equations





The Attempt at a Solution



[tex]let \ y = x^{\frac{1}{x}}[/tex]

[tex]\ln y = {\frac{1}{x}} ln x[/tex]

[tex]\lim_{x\rightarrow \infty } ln y = \lim_{x\rightarrow \infty } {\frac{ln x}{x}}[/tex]

L'H
[tex]\lim_{x\rightarrow \infty } ln y = \lim_{x\rightarrow \infty } {\frac{\frac{1}{x}}{1}} = \frac{0}{1} = 0[/tex]

Wolfram shows the answer to be 1 and intuition says that a^0 = 1 so i am not sure where my error is.

Why do you think you have made an error?
 
  • #4
eumyang
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[tex]let \ y = x^{\frac{1}{x}}[/tex]
It would be better to write it this way:
Let
[tex]let \ y = \lim_{x\rightarrow \infty } x^{\frac{1}{x}}[/tex]

[tex]\ln y = {\frac{1}{x}} \ln x[/tex]
When you take the natural logarithm of both sides, you actually get
[tex]\ln y = \ln \left( \lim_{x\rightarrow \infty } x^{\frac{1}{x}} \right)[/tex]
... but since ln x is continuous, you can rewrite it as
[tex]\ln y = \lim_{x\rightarrow \infty } \ln \left(x^{\frac{1}{x}} \right) = \lim_{x\rightarrow \infty } {\frac{1}{x}}\ln x[/tex]

[tex]\ln y = \lim_{x\rightarrow \infty } {\frac{\frac{1}{x}}{1}} = \frac{0}{1} = 0[/tex]
(EDIT: I removed the "lim" in front of ln y.)

Wolfram shows the answer to be 1 and intuition says that a^0 = 1 so i am not sure where my error is.
That's because you are not finished. You have one more step to go. What does that 0 represent?
 
  • #5
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That's because you are not finished. You have one more step to go. What does that 0 represent?

0 = ln y so e^0 = 1?
 
  • #6
eumyang
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Yes, ln y = 0, so y = e0 = 1. If you use my corrected definition of y, then you have your answer.
 
  • #8
HallsofIvy
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"a^0= 1" has nothing to do with this. If you just replace x with [itex]\infty[/itex], you get [itex]\infty^0[/itex] which is "undetermined".
 
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  • #10
Dick
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