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I'm doing some research for an algorithm, and I have a lot of curiosity, too. I'd like to learn how limits are calculated by mathematicians and mathematical software. I understand calculus somewhat, but I'm a litty rusty on it and I sometimes get perplexed by calculations involving limits. What's my best bet for learning about how limits are calculated? Should I review my calculus, or is there more science to limits (like numerical methods?!?)?

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On the numerical front, things get a little hazy... you'll want to look into the use of sequence transformation methods. Briefly, you construct an appropriate sequence for your limit; e.g. for something like $\lim\limits_{x\to 0} f(x)$, you might consider the sequence $f(x_k)$ where $x_k$ is a sequence that tends to 0 like $k^-n,\quad n>0$ or $c^k,\quad 0 < c < 1$, and then apply a transformation that accelerates the convergence. If that's what you're interested in, I'll write up something. –  J. M. Aug 25 '11 at 2:51
    
@J.M. Yes-I'm very interested in sequence transformation methods. Anything relating to limits is good, and this seems prime territory to explore! I'd like to thank you in advance for bringing this up. :-) –  Matt Groff Aug 25 '11 at 3:27
    
I've already given a short introduction in this answer (and also linked to one example where I used them here); if there's anything you still need that isn't there on sequence transformations, I'll post an elaboration. –  J. M. Aug 25 '11 at 3:29

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