Understanding Delta Epsilon Proofs: Recommended Calc Textbooks

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Can anyone recommend me a book that better explains delta epsilon proofs. We skimmed it in beginning calc class and I would like to understand it better.

Also I would like to know if there are any good books out there for learning calc. I do not like the stewart textbook we are using that much.



The folly of mistaking a paradox for a discovery, a metaphor for a proof, a torrent of verbiage for a string of capital truths, and ourselves for an oracle, is inborn in us- Paul Valerey
 
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I haven't read Stewart's Book, but I've only heard good things. A great book is "Calculus" by Michael Spivak, arguably the best calc book written to date! It has a wonderful explanation of the epsilon delta definition of limits. The exercises are probably more challenging than Stewarts though, so this book isn't for the faint of heart.
 
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Thanks for yuor reply. I will certainly give it a look!-Darius
 
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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