OpenMath Content Dictionary: limit1

Canonical URL:

http://www.openmath.org/cd/limit1.ocd

CD Base:

http://www.openmath.org/cd

CD File:

limit1.ocd

CD as XML Encoded OpenMath:

limit1.omcd

Defines:

above, below, both_sides, limit, null

Date:

2004-03-30

Version:

3 (Revision 1)

Review Date:

2006-03-30

Status:

official

---

     This document is distributed in the hope that it will be useful, 
     but WITHOUT ANY WARRANTY; without even the implied warranty of 
     MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

     The copyright holder grants you permission to redistribute this 
     document freely as a verbatim copy. Furthermore, the copyright
     holder permits you to develop any derived work from this document
     provided that the following conditions are met.
       a) The derived work acknowledges the fact that it is derived from
          this document, and maintains a prominent reference in the 
          work to the original source.
       b) The fact that the derived work is not the original OpenMath 
          document is stated prominently in the derived work.  Moreover if
          both this document and the derived work are Content Dictionaries
          then the derived work must include a different CDName element,
          chosen so that it cannot be confused with any works adopted by
          the OpenMath Society.  In particular, if there is a Content 
          Dictionary Group whose name is, for example, `math' containing
          Content Dictionaries named `math1', `math2' etc., then you should 
          not name a derived Content Dictionary `mathN' where N is an integer.
          However you are free to name it `private_mathN' or some such.  This
          is because the names `mathN' may be used by the OpenMath Society
          for future extensions.
       c) The derived work is distributed under terms that allow the
          compilation of derived works, but keep paragraphs a) and b)
          intact.  The simplest way to do this is to distribute the derived
          work under the OpenMath license, but this is not a requirement.
     If you have questions about this license please contact the OpenMath
     society at http://www.openmath.org.

  Author: OpenMath Consortium
  SourceURL: https://github.com/OpenMath/CDs
            

This CD holds the basic notion of the limits of unary functions whilst its variable tend (either from above, below or both sides) to a particular value.

---

limit

Role:

application

Description:

This symbol is used to denote the limit of a unary function. It takes 3 arguments: the limiting value of the argument, the method of approach (either null, above, below or both_sides) and the function.

Example:

The limit of sin(x) as x tends to 0

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="limit1" name="limit"/>
    <OMI> 0 </OMI>
    <OMS cd="limit1" name="both_sides"/>
    <OMBIND>
      <OMS cd="fns1" name="lambda"/>
        <OMBVAR>
        <OMV name="x"/>
        </OMBVAR>
        <OMA>
          <OMS cd="transc1" name="sin"/>
          <OMV name="x"/>
        </OMA>
    </OMBIND>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="limit1">limit</csymbol>
  <cn type="integer">0</cn>
  <csymbol cd="limit1">both_sides</csymbol>
  <bind><csymbol cd="fns1">lambda</csymbol>
   <bvar><ci>x</ci></bvar>
   <apply><csymbol cd="transc1">sin</csymbol><ci>x</ci></apply>
  </bind>
 </apply>
</math>

Prefix

limit ( 0 , both_sides, lambda [ x ] . (sin ( x) ) )

Popcorn

limit1.limit(0, limit1.both_sides, fns1.lambda[$x -> sin($x)])

Rendered Presentation MathML

$$\underset{x\to 0}{\mathrm{limit}}\sin \left(x\right)$$

Signatures:

sts

---

[Next: both_sides] [Last: null] [Top]

---

both_sides

Role:

constant

Description:

This symbol is used within a limit construct to show the limit is being approached from both sides. It takes no arguments.

Example:

The limit of sin(x) as x tends to 0 from either direction

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="limit1" name="limit"/>
    <OMI> 0 </OMI>
    <OMS cd="limit1" name="both_sides"/>
    <OMBIND>
      <OMS cd="fns1" name="lambda"/>
        <OMBVAR>
        <OMV name="x"/>
        </OMBVAR>
        <OMA>
          <OMS cd="transc1" name="sin"/>
          <OMV name="x"/>
        </OMA>
    </OMBIND>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="limit1">limit</csymbol>
  <cn type="integer">0</cn>
  <csymbol cd="limit1">both_sides</csymbol>
  <bind><csymbol cd="fns1">lambda</csymbol>
   <bvar><ci>x</ci></bvar>
   <apply><csymbol cd="transc1">sin</csymbol><ci>x</ci></apply>
  </bind>
 </apply>
</math>

Prefix

limit ( 0 , both_sides, lambda [ x ] . (sin ( x) ) )

Popcorn

limit1.limit(0, limit1.both_sides, fns1.lambda[$x -> sin($x)])

Rendered Presentation MathML

$$\underset{x\to 0}{\mathrm{limit}}\sin \left(x\right)$$

Signatures:

sts

---

[Next: above] [Previous: limit] [Top]

---

above

Role:

constant

Description:

This symbol is used within a limit construct to show the limit is being approached from above. It takes no arguments.

Example:

The limit of sin(x) as x tends to 0 from above

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="limit1" name="limit"/>
    <OMI> 0 </OMI>
    <OMS cd="limit1" name="above"/>
    <OMBIND>
      <OMS cd="fns1" name="lambda"/>
        <OMBVAR>
        <OMV name="x"/>
        </OMBVAR>
        <OMA>
          <OMS cd="transc1" name="sin"/>
          <OMV name="x"/>
        </OMA>
    </OMBIND>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="limit1">limit</csymbol>
  <cn type="integer">0</cn>
  <csymbol cd="limit1">above</csymbol>
  <bind><csymbol cd="fns1">lambda</csymbol>
   <bvar><ci>x</ci></bvar>
   <apply><csymbol cd="transc1">sin</csymbol><ci>x</ci></apply>
  </bind>
 </apply>
</math>

Prefix

limit ( 0 , above, lambda [ x ] . (sin ( x) ) )

Popcorn

limit1.limit(0, limit1.above, fns1.lambda[$x -> sin($x)])

Rendered Presentation MathML

$$\underset{x\to 0^{+}}{\mathrm{limit}}\sin \left(x\right)$$

Signatures:

sts

---

[Next: below] [Previous: both_sides] [Top]

---

below

Role:

constant

Description:

This symbol is used within a limit construct to show the limit is being approached from below. It takes no arguments.

Example:

The limit of sin(x) as x tends to 0 from below

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="limit1" name="limit"/>
    <OMI> 0 </OMI>
    <OMS cd="limit1" name="below"/>
    <OMBIND>
      <OMS cd="fns1" name="lambda"/>
        <OMBVAR>
        <OMV name="x"/>
        </OMBVAR>
        <OMA>
          <OMS cd="transc1" name="sin"/>
          <OMV name="x"/>
        </OMA>
    </OMBIND>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="limit1">limit</csymbol>
  <cn type="integer">0</cn>
  <csymbol cd="limit1">below</csymbol>
  <bind><csymbol cd="fns1">lambda</csymbol>
   <bvar><ci>x</ci></bvar>
   <apply><csymbol cd="transc1">sin</csymbol><ci>x</ci></apply>
  </bind>
 </apply>
</math>

Prefix

limit ( 0 , below, lambda [ x ] . (sin ( x) ) )

Popcorn

limit1.limit(0, limit1.below, fns1.lambda[$x -> sin($x)])

Rendered Presentation MathML

$$\underset{x\to 0_{-}}{\mathrm{limit}}\sin \left(x\right)$$

Signatures:

sts

---

[Next: null] [Previous: above] [Top]

---

null

Role:

constant

Description:

This symbol is used within a limit construct to avoid specifying the method of approach to the limit. It takes no arguments.

Example:

The limit of sin(x) as x tends to 0

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="limit1" name="limit"/>
    <OMI> 0 </OMI>
    <OMS cd="limit1" name="null"/>
    <OMBIND>
      <OMS cd="fns1" name="lambda"/>
      <OMBVAR>
        <OMV name="x"/>
      </OMBVAR>
      <OMA>
        <OMS cd="transc1" name="sin"/>
        <OMV name="x"/>
      </OMA>
    </OMBIND>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="limit1">limit</csymbol>
  <cn type="integer">0</cn>
  <csymbol cd="limit1">null</csymbol>
  <bind><csymbol cd="fns1">lambda</csymbol>
   <bvar><ci>x</ci></bvar>
   <apply><csymbol cd="transc1">sin</csymbol><ci>x</ci></apply>
  </bind>
 </apply>
</math>

Prefix

limit ( 0 , null, lambda [ x ] . (sin ( x) ) )

Popcorn

limit1.limit(0, limit1.null, fns1.lambda[$x -> sin($x)])

Rendered Presentation MathML

$$\underset{x\to 0}{\mathrm{limit}}\sin \left(x\right)$$

Signatures:

sts

---

[First: limit] [Previous: below] [Top]