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To be honest, I nearly don't understand a thing. What's going on, and when am I done with the proof?

I can plug in the limits and the approached value into the formal definiton of a limit, but that's as far as I get. (I semi-get the easy ones such as Lim(x->1)(3x+1)=4)

So I wonder: Could any of you write an explanation (with a few easy and advanced examples), and feed it all to me with a teaspoon? Can you please explain every single step you make? If you can do it in words in addition to mathematical expressions I would be very grateful. A problem I have is not seeing the connection between Epsilon and Delta, so a baby's description in plain english would help (not that we "easily" see that blablabla imples blabla :P)

And can you please explain how and why you pick certain values for epsilon and delta? I don't understand a thing of what it says in my solution guide (Robert A. Adams)...

I'm going to philosophy class now, but I really won't to understand this thourougly :)

And at the end: are we supposed do use this teqniue for evaluating lim(x->infinity)sinx ? Maybe I'll see it after I've "processed" your replys (if you make any)

Thanks alot!

Cheers!