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# Limit

 Limit[expr, x->x0]finds the limiting value of expr when x approaches x0.
• Limit[expr, x->x0, Direction->1] computes the limit as x approaches x0 from smaller values. Limit[expr, x->x0, Direction->-1] computes the limit as x approaches x0 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[expr, x->x0] uses the setting , which determines the direction from assumptions that have been given, using Direction->-1 as the default. For limit points at infinity, the direction is determined from the direction of the infinity.
 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 limits along coordinate axes are zero:
The limit along the diagonal:
Behavior around the origin:
Many different values can be achieved:
Differentiation by integration: