What are LimInf and LimSup and how do they relate to irrational numbers?

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Disregard, figured it out.
 
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The first three are pretty easy in terms of epsilon-delta, if you really want to go that way. (d) isn't. You need to know pi is irrational and what that implies about the integers mod pi. Though it is sort of clear 'intuitively'. This makes me think the question is more about understanding what lim inf and lim sup are, rather than about proving what they are. And I think you do understand that.
 

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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