Left and Right Hand Limits Exist in Real Analysis: A Tutorial

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This is a simple question. I need to know if I'm reading the notation correctly.

"Show that for each c in (a,b) the limit f(x) exists as x --> c- and as x --> c+."

This is in Royden's Real Analysis.

What I'm going to do is show that the left and right hand limits exist... right? Just like in Calc. I 15 years ago... like I remember any of this... LOL.

But that's what I'm supposed to do here? Correct?
 
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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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