How Do You Calculate the Limit of a Factorial?

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Please can anybody help me find this limit?
 

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It's hard to believe you are serious. Have you done anything on this yourself? Have you tried, for example, calculating values for, say, x= 10, 1000, etc.?
 
I studied the limit but not this type

this question is older than me :)
 
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Just a few things that should have made this limit quite obvious -

n! < n^n, n > 2

n^2 > n

Perhaps you have seen how to evaluate the common \lim_{n\to \infty} n^{\frac{1}{n}}?
 
Gib Z: Tanks a lot.
 

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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