limit
Compute a limit
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limit(f
,x
, <Left  Right  Real
>, <Intervals>, <NoWarning>) limit(f
,x = x_{0}
, <Left  Right  Real
>, <Intervals>, <NoWarning>)
limit(f, x = x_{0}, Real)
computes
the bidirectional limit , .
limit(f, x = x_{0}, Left  Right)
computes
the onesided limit , respectively.
limit(f, x = x_{0}, Intervals)
computes
a set containing all accumulation points of , .
limit(f, x = x0, <Real>)
computes the
bidirectional limit of f
when x
tends
to x0
on the real axis. The limit point x0
may
be omitted, in which case limit
assumes x0
= 0
.
If the limit point x0
is infinity or 
∞, then the limit is taken from the left
to infinity or
from the right to  ∞,
respectively.
If provably no limit exists, then undefined
is returned. See Example 2.
limit(f, x = x0, Left)
returns the limit
when x
tends to x0
from the
left. limit(f, x = x0, Right)
returns the limit
when x
tends to x0
from the
right. See Example 2.
If it cannot be determined whether a limit exist, or cannot
determine its value, then a symbolic to limit
is
returned. See Example 3. The same holds,
in case the option Intervals
is given, if no information
on the set of accumulation points could be obtained.
If f
contains parameters, then limit
reacts
to properties of
those parameters set by assume
.
See Example 5. It may also return a
case analysis (piecewise
)
depending on these parameters.
You can compute the limit of a piecewise function. The conditions you use to define a piecewise function can depend on the limit variable. See Example 6.
Internally, limit
tries to determine the
limit from a series expansion of f
around x
= x0
computed via series
.
It may be necessary to increase the value of the environment variable ORDER
in
order to find the limit.
limit
works on a symbolic level and should
not be called with arguments containing floating
point arguments.
The function is sensitive to the environment variable ORDER
,
which determines the default number of terms in series computations
(see series
).
Properties of
identifiers set by assume
are
taken into account.
The following command computes :
limit((1  cos(x))/x^2, x)
A possible definition of e is given by the limit of the sequence for :
limit((1 + 1/n)^n, n = infinity)
Here is a more complex example:
limit( (exp(x*exp(x)/(exp(x) + exp(2*x^2/(x+1))))  exp(x))/x, x = infinity )
The bidirectional limit of for does not exist:
limit(1/x, x = 0)
You can compute the onesided limits from the left and from
the right by passing the options Left
and Right
,
respectively:
limit(1/x, x = 0, Left), limit(1/x, x = 0, Right)
If limit
is not able to compute the limit,
then a symbolic limit
call is returned:
delete f: limit(f(x), x = infinity)
The function sin(x) oscillates for between  1 and 1; no accumulation points outside that interval exist:
limit(sin(x), x = infinity, Intervals)
In fact, all elements of the interval returned are accumulation points. This need not be the case in general. In the following example, the limit inferior and the limit superior are in fact and , respectively:
limit(sin(1/x) + cos(1/x), x = 0, Intervals)
limit
is not able to compute the limit of x^{n} for without
additional information about the parameter n:
assume(n in R_): limit(x^n, x = infinity)
We can also assume
immediately
that n > 0 and
get no case analysis then:
assume(n > 0): limit(x^n, x = infinity)
Similarly, we can assume that n < 0:
assume(n < 0): limit(x^n, x = infinity)
delete n:
Compute limit of the piecewise function:
limit(piecewise([x^3 > 10000*x, 1/x], [x^3 <= 10000*x, 10]), x = infinity)
Compute limits of the incomplete Gamma function:
limit(igamma(z, t), t = infinity); limit(igamma(z, t), t = 0)

An arithmetical
expression representing a function in 

An identifier 

The limit point: an arithmetical expression, possibly 

This controls the direction of the limit computation. The option 

Either 

If this option is set to 
arithmetical
expression. If the option Intervals
was
given, the result is a (finite or infinite) set.
f
limit
uses an algorithm based on the thesis
of Dominik Gruntz: “On Computing Limits in a Symbolic Manipulation
System”, Swiss Federal Institute of Technology, Zurich, Switzerland,
1995. If this fails, it tries to proceed recursively; finally, it
attempts a series expansion.