What is the Limit of t raised to the Power of 1/t as t Approaches Infinity?

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Homework Statement



\lim_{t\rightarrow \infty } t^{\frac{1}{t}}

Homework Equations


The Attempt at a Solution



let \ y = x^{\frac{1}{x}}

\ln y = {\frac{1}{x}} ln x

\lim_{x\rightarrow \infty } ln y = \lim_{x\rightarrow \infty } {\frac{ln x}{x}}

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

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

Homework Statement



\lim_{t\rightarrow \infty } t^{\frac{1}{t}}

Homework Equations





The Attempt at a Solution



let \ y = x^{\frac{1}{x}}

\ln y = {\frac{1}{x}} ln x

\lim_{x\rightarrow \infty } ln y = \lim_{x\rightarrow \infty } {\frac{ln x}{x}}

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

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?
 
freezer said:
let \ y = x^{\frac{1}{x}}
It would be better to write it this way:
Let
let \ y = \lim_{x\rightarrow \infty } x^{\frac{1}{x}}

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

freezer said:
\ln y = \lim_{x\rightarrow \infty } {\frac{\frac{1}{x}}{1}} = \frac{0}{1} = 0
(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?
 
eumyang said:
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?
 
Yes, ln y = 0, so y = e0 = 1. If you use my corrected definition of y, then you have your answer.
 
"a^0= 1" has nothing to do with this. If you just replace x with \infty, you get \infty^0 which is "undetermined".
 
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