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

# Limit

 Limitfinds the limiting value of expr when x approaches .
• Limit[expr, x->x0, Direction->1] computes the limit as x approaches from smaller values. Limit[expr, x->x0, Direction->-1] computes the limit as x approaches from larger values.
• Limit returns Interval objects to represent ranges of possible values, for example at essential singularities.
• Limit returns unevaluated when it encounters functions about which it has no specific information. Limit therefore by default makes no explicit assumptions about symbolic functions.
• Assumptions can be specified as a setting for the option Assumptions.
• Limit uses the setting Direction, which determines the direction from assumptions that have been given, using Direction as the default. For limit points at infinity, the direction is determined from the direction of the infinity.
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 Scope   (6)
Limits of rational functions:
Algebraic functions:
Piecewise functions:
Elementary functions:
Special functions:
Find a limit of a bounded oscillating function at essential singularity:
 Options   (7)
Default settings assume generic functions are not analytic:
Assume analyticity:
The limit may not exist or be different for different parameter values:
Parameter-dependent limit:
Limits from the left and right:
Limits at piecewise discontinuities:
Limits at a pole:
Limits at a branch cut:
 Applications   (5)
Find a linear asymptote:
Riemann sums:
Improper integrals:
Check the asymptotic complexity of the fast Fourier transform:
Construct a rotation matrix as a limit of repeated infinitesimal transformations:
Difference quotients:
The :
The limits along coordinate axes are zero:
The limit along the diagonal:
Behavior around the origin:
Many different values can be achieved:
Limit may return an incorrect answer for an inexact input:
The result is correct when an exact input is used:
Numerical cancellations are behind the incorrect result:
Differentiation by integration: