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

"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?

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

do you mean $\displaystyle lim_{x \to \infty} inf(x^{1/x}) $ = 1 ?

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

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**phys251** "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 $\displaystyle \lim _{x \to \infty } \frac{{\ln (x)}}{x} = ?$.

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

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Originally Posted by

**jakncoke** do you mean $\displaystyle lim_{x \to \infty} inf(x^{1/x}) $ = 1 ?

Yes.

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Originally Posted by

**Plato** 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?

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

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

Quote:

Originally Posted by

**phys251** That limit would be zero. So I would just take e^0=1 and be done?

Yup, it indeed was that simple. Thanks, Plato!