What is the limit of the expression (3sqrt{n})^(1/2n)?

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Homework Statement


Determine the limit of:

lim ((3sqrt{n})^(1/2n))


Homework Equations





The Attempt at a Solution


I don't even know where to begin...perhaps squaring the entire term so I get..

9n^(1/[4n^2]) which is equivalent to n^[1/(n^2)]

But I don't know what the limit of that is...I graph it and get all kinds of craziness
Please help!
 
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The limit as n -> ?
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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