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http://functions.wolfram.com/04.07.04.0005.01
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Limit[Quotient[m + \[Epsilon], n], \[Epsilon] -> Plus[0]] ==
Quotient[m, n] /; Element[Re[m/n], Integers] && Element[n, Reals] && n > 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Limit", "[", RowBox[List[RowBox[List["Quotient", "[", RowBox[List[RowBox[List["m", "+", "\[Epsilon]"]], ",", "n"]], "]"]], ",", RowBox[List["\[Epsilon]", "\[Rule]", RowBox[List["+", "0"]]]]]], "]"]], "\[Equal]", RowBox[List["Quotient", "[", RowBox[List["m", ",", "n"]], "]"]]]], "/;", RowBox[List[RowBox[List["Element", "[", RowBox[List[RowBox[List["Re", "[", FractionBox["m", "n"], "]"]], ",", "Integers"]], "]"]], "\[And]", RowBox[List["n", "\[Element]", "Reals"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <munder> <mi> lim </mi> <mrow> <mi> ϵ </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mrow> <mo> + </mo> <mn> 0 </mn> </mrow> </mrow> </munder> <mo> ⁢ </mo> <mtext>   </mtext> <mrow> <mi> quotient </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> m </mi> <mo> + </mo> <mi> ϵ </mi> </mrow> <mo> , </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mi> quotient </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mi> m </mi> <mi> n </mi> </mfrac> <mo> ) </mo> </mrow> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[Reals]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> n </mi> <mo> > </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <limit /> <bvar> <ci> ϵ </ci> </bvar> <condition> <apply> <tendsto /> <ci> ϵ </ci> <apply> <plus /> <cn type='integer'> 0 </cn> </apply> </apply> </condition> <apply> <ci> quotient </ci> <apply> <plus /> <ci> m </ci> <ci> ϵ </ci> </apply> <ci> n </ci> </apply> </apply> <apply> <ci> quotient </ci> <ci> m </ci> <ci> n </ci> </apply> </apply> <apply> <and /> <apply> <in /> <apply> <real /> <apply> <times /> <ci> m </ci> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <integers /> </apply> <apply> <in /> <ci> n </ci> <reals /> </apply> <apply> <gt /> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Limit", "[", RowBox[List[RowBox[List["Quotient", "[", RowBox[List[RowBox[List["m_", "+", "\[Epsilon]_"]], ",", "n_"]], "]"]], ",", RowBox[List["\[Epsilon]_", "\[Rule]", RowBox[List["+", "0"]]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Quotient", "[", RowBox[List["m", ",", "n"]], "]"]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", FractionBox["m", "n"], "]"]], "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[Element]", "Reals"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
