This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.
 NUMERICAL CALCULUS PACKAGE SYMBOL

# NLimit

 numerically finds the limiting value of expr as z approaches .
• The expression expr must be numeric when its argument z is numeric.
• constructs a sequence of values that approach the point and uses extrapolation to find the limit.
• is unable to recognize small numbers that should in fact be zero. Chop may be needed to eliminate these spurious residuals.
• often fails when the limit has a power law approach to infinity.
• The following options can be given:
 WorkingPrecision MachinePrecision precision to use in internal computations Direction Automatic vector giving the direction of approach Scale 1 initial step size in the sequence of steps Terms 7 number of terms used to evaluate the limit Method EulerSum the method used to evaluate the result WynnDegree 1 degree used in Wynn's epsilon algorithm
• The option Direction->d specifies that the approach vector to a finite limit point is given by the complex number d. The default setting Direction is equivalent to Direction, and computes the limit as z approaches from larger values.
• approaches infinite limit points on a ray from the origin.
• The option Scale specifies the initial step in the constructed sequence.
• For finite limit points , the initial step is a distance Scale away from . For infinite limit points, the initial step is a distance Scale away from the origin.
• The accuracy of the result is generally improved by increasing the number of terms, although increased WorkingPrecision will also usually be necessary.
• Possible settings for Method include:
 EulerSum converts sequence to a sum and uses EulerSum SequenceLimit uses on constructed sequence
• The option specifies the number of iterations of Wynn's epsilon algorithm to be used by . In general, there must be at least terms for iterations.
Find the limit at zero:
Find the limit at infinity:
Needs["NumericalCalculus`"]
Find the limit at zero:
 Out[2]=

Needs["NumericalCalculus`"]
Find the limit at infinity:
 Out[2]=
 Scope   (2)
The expression can be manifestly complex:
The limit point can be complex:
 Options   (8)
Expressions which approach their limiting value exponentially need fewer terms:
Increasing the number of terms can improve accuracy:
Error in numerical approximation:
Use more terms to reduce error:
Use Scale to avoid regions where the expression is undefined:
The function diverges for , so choose the initial step to avoid this divergence:
Approach 0 along the negative real axis ( can be input using zEsc co Esc):
Approach 0 along the positive imaginary axis:
Approach 0 from the quadrant, 225°:
An example where the default method works fairly well:
Using produces poorer results:
An example where the default method works poorly:
Here, produces the correct result:
When using Method, increasing may improve the accuracy of the limit:
Error with :
Error with :
Increasing WorkingPrecision alone does not produce a more accurate result:
Error with WorkingPrecision:
Error with WorkingPrecision:
To improve accuracy, the number of terms needs to be increased:
 Applications   (2)
Find the limit of a numerically defined function:
Limits where parts of the expression have essential singularities:
In this case, the exact limit can be found:
Check:
can be used directly to compute limits:
Limits whose value approaches infinity are sometimes unable to be computed: