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Residues

Limit tells you what the value of expr is when x tends to . When this value is infinite, it is often useful instead to know the residue of expr when x equals . The residue is given by the coefficient of in the power series expansion of expr about the point .
Residue[expr,{x,x0}]the residue of expr when x equals

Computing residues.

The residue here is equal to 1.
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The residue here is zero.
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Residues can be computed at the point at infinity.
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