Prove that lim x->inf (x ^ (1/x)) = 1

Jun 2012
220
20
Georgia
"Prove that lim x->inf (x ^ (1/x)) = 1."

I'm stuck on this. I was able to show that the limit exists, is finite, and therefore unique. But showing that it equals 1 is eluding me. If I raise both sides to the xth power, I get x = 1^x, which is clearly wrong. So what do I do to show that the limit is 1? Proof by contradiction? Induction? Any hints on how to proceed?
 
May 2010
387
93
do you mean \(\displaystyle lim_{x \to \infty} inf(x^{1/x}) \) = 1 ?
 

Plato

MHF Helper
Aug 2006
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8,653
"Prove that lim x->inf (x ^ (1/x)) = 1.

You know that if \(\displaystyle y=x^{f(x)}\) then \(\displaystyle y=e^{f(x)\ln(x)\).

What is \(\displaystyle \lim _{x \to \infty } \frac{{\ln (x)}}{x} = ?\).
 
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Jun 2012
220
20
Georgia
do you mean \(\displaystyle lim_{x \to \infty} inf(x^{1/x}) \) = 1 ?
Yes.

You know that if \(\displaystyle y=x^{f(x)}\) then \(\displaystyle y=e^{f(x)\ln(x)\).

What is \(\displaystyle \lim _{x \to \infty } \frac{{\ln (x)}}{x} = ?\).
That limit would be zero. So I would just take e^0=1 and be done?