Next: Differentiation, Previous: Elliptic Functions [Contents][Index]
Default value: 4
lhospitallim is the maximum number of times L’Hospital’s
rule is used in limit. This prevents infinite looping in cases like
limit (cot(x)/csc(x), x, 0).
Computes the limit of expr as the real variable x approaches the
value val from the direction dir. dir may have the value
plus for a limit from above, minus for a limit from below, or
may be omitted (implying a two-sided limit is to be computed).
limit uses the following special symbols: inf (positive infinity)
and minf (negative infinity). On output it may also use und
(undefined), ind (indefinite but bounded) and infinity (complex
infinity).
infinity (complex infinity) is returned when the limit of
the absolute value of the expression is positive infinity, but
the limit of the expression itself is not positive infinity or
negative infinity. This includes cases where the limit of the
complex argument is a constant, as in limit(log(x), x, minf),
cases where the complex argument oscillates, as in
limit((-2)^x, x, inf), and cases where the complex
argument is different for either side of a two-sided limit, as in
limit(1/x, x, 0) and limit(log(x), x, 0).
lhospitallim is the maximum number of times L’Hospital’s rule
is used in limit. This prevents infinite looping in cases like
limit (cot(x)/csc(x), x, 0).
tlimswitch when true will allow the limit command to use
Taylor series expansion when necessary.
limsubst prevents limit from attempting substitutions on
unknown forms. This is to avoid bugs like limit (f(n)/f(n+1), n, inf)
giving 1. Setting limsubst to true will allow such
substitutions.
limit with one argument is often called upon to simplify constant
expressions, for example, limit (inf-1).
example (limit) displays some examples.
For the method see Wang, P., "Evaluation of Definite Integrals by Symbolic Manipulation", Ph.D. thesis, MAC TR-92, October 1971.
Default value: false
prevents limit from attempting substitutions on unknown forms. This is
to avoid bugs like limit (f(n)/f(n+1), n, inf) giving 1. Setting
limsubst to true will allow such substitutions.
Take the limit of the Taylor series expansion of expr in x
at val from direction dir.
Default value: true
When tlimswitch is true, the limit command will use a
Taylor series expansion if the limit of the input expression cannot be computed
directly. This allows evaluation of limits such as
limit(x/(x-1)-1/log(x),x,1,plus). When tlimswitch is false
and the limit of input expression cannot be computed directly, limit will
return an unevaluated limit expression.
Compute limit of expression expr with respect to variable var at value.
When value is not infinite (i.e., not inf or minf),
direction must be supplied,
either plus for a limit from above,
or minus for a limit from below.
If gruntz cannot find the limit,
an unevaluated expression gruntz(...) is returned.
gruntz implements the method described in the dissertation of
Dominik Gruntz, "On Computing Limits in a Symbolic Manipulation System"
(ETH Zurich, 1996).
The algorithm identifies the most rapidly varying subexpression, replaces it with a new variable, rewrites the expression in terms of the new variable, and then repeats.
The algorithm doesn’t handle oscillating functions, so it can’t do things like
limit(sin(x)/x, x, inf).
To handle limits involving functions such as gamma(x) and erf(x),
the Gruntz algorithm requires them to be written in terms of asymptotic expansions,
which Maxima cannot currently do.
The Gruntz algorithm assumes that variables and expressions are real,
so, for example, it can’t handle limit((-2)^x, x, inf).
gruntz is one of the methods called from limit.
Next: Differentiation, Previous: Elliptic Functions [Contents][Index]