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Gruntz

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Description

Summary

Gruntz's algorithm for the limit of an expression as x tends to positive infinity,
after D. Gruntz, "On Computing Limits in a Symbolic Manipulation System", ETH 1996.

Remarks

The idea is to work out which subexpressions grow fastest, rewrite the expression in
terms of a single one of them, and read the answer off the leading term of the power
series that leaves. Two functions are in the same comparability class when
log|f| / log|g| tends to something finite and non-zero, and the set of subexpressions
in the fastest class present is the mrv set. Every member of it can be written as a
power of any other member times something slower, so one member is picked -- turned
upside down if need be, so that it tends to zero -- and everything is written in terms
of it. What is left is a series in that one quantity, and the sign of the leading
exponent says whether the limit is zero, infinite, or the limit of the leading
coefficient, which is a smaller problem of the same kind.
 This answers the limits that defeat term-by-term expansion because the terms cancel to
every order. lim x -> +oo e^(x + e^(-x)) - e^x is the standard one: expanding the two
exponentials separately gives two divergent series whose difference cancels entirely,
while rewriting the whole expression in w = e^(-x) gives (e^w - 1)/w, whose leading
term is 1.
 The scope here is the exp-log functions: what can be built from x and the rationals
with the four operations, exp and log. That is the class the algorithm is proved for,
because those functions are eventually monotone and so comparable at all. Anything
else -- a sine, an unknown function, a factorial -- makes this decline rather than
guess, since sin(x) has no limit at infinity and no comparability class either.

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