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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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    [l'Hôpital's rule](http://en.wikipedia.org/wiki/L'H%C3%B4pital's_rule), [Taylor series](http://en.wikipedia.org/wiki/Taylor_series), [Big O notation](http://en.wikipedia.org/wiki/Big_O_notation), [convergence tests](http://en.wikipedia.org/wiki/Convergence_tests), [telescoping series](http://en.wikipedia.org/wiki/Telescoping_series), [series acceleration](http://en.wikipedia.org/wiki/Series_acceleration), ...2011-08-25
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    http://www.khanacademy.org/search?page_search_query=calculus2011-08-25
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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.2011-08-25
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    @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. :-)2011-08-25
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    I've already given a short introduction in [this answer](http://math.stackexchange.com/questions/47272/53407#53407) (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.2011-08-25
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    @J.M. Thanks again. I'm kind of overwhelmed at the wealth of information and knowledge. I kind of don't know what to say, other than, wow!2011-08-25

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