Solving Homework Problem: ln and e Inverse Process

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Homework Statement



http://img513.imageshack.us/img513/2687/problem15001.jpg

Homework Equations



I know the answer is e but i forgat how to do the lny=xln(1+1/n) and then I solve an get an e because ln and e are inverse but i forgot the process can you please show me the process

The Attempt at a Solution

 
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at the stage

\stackrel{lim}{n \rightarrow \infty}, ln(y) = n.ln(1+1/n)

the limit is indeterminant 0 * infinity, so can you re-write it and use L'Hopitals rule...?
 

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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