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

#### phys251

"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?

#### jakncoke

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

#### Plato

MHF Helper
"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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#### phys251

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?

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#### 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!