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

• Feb 13th 2013, 02:21 PM
phys251
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?
• Feb 13th 2013, 02:26 PM
jakncoke
Re: Prove that lim x->inf (x ^ (1/x)) = 1
do you mean $lim_{x \to \infty} inf(x^{1/x})$ = 1 ?
• Feb 13th 2013, 02:28 PM
Plato
Re: Prove that lim x->inf (x ^ (1/x)) = 1
Quote:

Originally Posted by phys251
"Prove that lim x->inf (x ^ (1/x)) = 1.

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

What is $\lim _{x \to \infty } \frac{{\ln (x)}}{x} = ?$.
• Feb 13th 2013, 02:29 PM
phys251
Re: Prove that lim x->inf (x ^ (1/x)) = 1
Quote:

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

Yes.

Quote:

Originally Posted by Plato
You know that if $y=x^{f(x)}$ then $y=e^{f(x)\ln(x)$.

What is $\lim _{x \to \infty } \frac{{\ln (x)}}{x} = ?$.

That limit would be zero. So I would just take e^0=1 and be done?
• Feb 13th 2013, 02:37 PM
jakncoke
Re: Prove that lim x->inf (x ^ (1/x)) = 1
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• Feb 13th 2013, 03:36 PM
phys251
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!