Solving a limit with L'H's rule

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Homework Statement



OErDOT2.png


Homework Equations


The Attempt at a Solution



YG1M1Fh.png


Are you allowed to do that where you just apply L'H an infinite amount of times?
 
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Turion said:

Homework Statement



OErDOT2.png


Homework Equations





The Attempt at a Solution



YG1M1Fh.png

After taking the derivative of the denominator, you should get et, not eα. I suspect that was a typo.

Anyway, your limit is correct. The basic idea is that power functions, such as tα grow large, but exponential functions grow large faster.
 

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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