Taking the Limit As N -> Infinity

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Taking the Limit As N --> Infinity

\mathop {\lim }\limits_{n \to \infty } \frac{{n^n x}}{{(n + 1)^n }}

Does this limit exist? Somehow it's supposed to come down to x/e
 
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Factor out the x (the limit is with respect to n).

Consolidate the remaining factors into a term with a single exponent.

Perform a little magic inside the parentheses and look for something familiar.
 
Got everything except the magic... the goal is to get (1+(1/n))^n, right?
 
Well, that would be e. You want 1/e.
 
Doh! Thanks. Got it.
 

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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