User:IssaRice/Tao's notation for limits - Revision history

https://machinelearning.subwiki.org/w/index.php?action=history&feed=atom&title=User%3AIssaRice%2FTao%27s_notation_for_limits User:IssaRice/Tao's notation for limits - Revision history 2026-05-26T05:45:08Z Revision history for this page on the wiki MediaWiki 1.41.2 https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Tao%27s_notation_for_limits&diff=1019&oldid=prev IssaRice at 03:23, 1 December 2018 2018-12-01T03:23:02Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 03:23, 1 December 2018</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l25">Line 25:</td> <td colspan="2" class="diff-lineno">Line 25:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| Tao's notation || <math>\lim_{x\to x_0;\, x\in E} f(x)</math> || Works with anonymous functions. || Somewhat verbose: the "<math>x\in E</math>" part must be written out each time.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| Tao's notation || <math>\lim_{x\to x_0;\, x\in E} f(x)</math> || Works with anonymous functions. || Somewhat verbose: the "<math>x\in E</math>" part must be written out each time.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| Function restriction notation || <math>\lim_{x\to x_0} f|_E(x)</math> || || Doesn't work with anonymous functions.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| Function restriction notation || <math>\lim_{x\to x_0} f|_E(x)</math> || <ins style="font-weight: bold; text-decoration: none;">Consistent with the standard notation. </ins>|| Doesn't work with anonymous functions.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| Standard notation || <math>\lim_{x\to x_0} f(x)</math> || Works with anonymous functions. || Set used in the limit does not appear in the notation, so one must assume some default set (such as the entire domain of the function) or introduce new notation for e.g. one-sided limits as e.g. <math>\lim_{x\to x_0+} f(x)</math> and <math>\lim_{x\to x_0-} f(x)</math>.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| Standard notation || <math>\lim_{x\to x_0} f(x)</math> || Works with anonymous functions. || Set used in the limit does not appear in the notation, so one must assume some default set (such as the entire domain of the function) or introduce new notation for e.g. one-sided limits as e.g. <math>\lim_{x\to x_0+} f(x)</math> and <math>\lim_{x\to x_0-} f(x)</math>.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|}</div></td></tr> </table>

IssaRice https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Tao%27s_notation_for_limits&diff=1018&oldid=prev IssaRice at 03:21, 1 December 2018 2018-12-01T03:21:38Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 03:21, 1 December 2018</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l27">Line 27:</td> <td colspan="2" class="diff-lineno">Line 27:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| Function restriction notation || <math>\lim_{x\to x_0} f|_E(x)</math> || || Doesn't work with anonymous functions.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| Function restriction notation || <math>\lim_{x\to x_0} f|_E(x)</math> || || Doesn't work with anonymous functions.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| Standard notation || <math>\lim_{x\to x_0} f(x)</math> || Works with anonymous functions. ||</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| Standard notation || <math>\lim_{x\to x_0} f(x)</math> || Works with anonymous functions. || <ins style="font-weight: bold; text-decoration: none;">Set used in the limit does not appear in the notation, so one must assume some default set (such as the entire domain of the function) or introduce new notation for e.g. one-sided limits as e.g. <math>\lim_{x\to x_0+} f(x)</math> and <math>\lim_{x\to x_0-} f(x)</math>.</ins></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|}</div></td></tr> </table>

IssaRice https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Tao%27s_notation_for_limits&diff=1017&oldid=prev IssaRice at 03:19, 1 December 2018 2018-12-01T03:19:50Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 03:19, 1 December 2018</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l13">Line 13:</td> <td colspan="2" class="diff-lineno">Line 13:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Rather than thinking of this as a result per se, I think it's better to think of this as ''alternative notations''. Why might one prefer one notation over the other? I think the strength of Tao's notation is that it works for anonymous functions/expressions. To be able to use the function restriction notation <math>f|_E</math>, we must have named our function beforehand. To give an example, we can write something like <math>\lim_{x\to 0;\, x\in (0,\infty)} |x|/x = 1</math>, but this is difficult to write in the other notation; we would have to say something like, "Let <math>f:\mathbf R\setminus \{0\} \to \mathbf R</math> be defined by <math>f(x) := |x|/x</math>. Then we have <math>\lim_{x\to 0} f|_{(0,\infty)}(x) = 1</math>."</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Rather than thinking of this as a result per se, I think it's better to think of this as ''alternative notations''. Why might one prefer one notation over the other? I think the strength of Tao's notation is that it works for anonymous functions/expressions. To be able to use the function restriction notation <math>f|_E</math>, we must have named our function beforehand. To give an example, we can write something like <math>\lim_{x\to 0;\, x\in (0,\infty)} |x|/x = 1</math>, but this is difficult to write in the other notation; we would have to say something like, "Let <math>f:\mathbf R\setminus \{0\} \to \mathbf R</math> be defined by <math>f(x) := |x|/x</math>. Then we have <math>\lim_{x\to 0} f|_{(0,\infty)}(x) = 1</math>."</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>An amusing fact/potential advantage of the function restriction notation is that one can write it in a ''point-free notation''. Namely, we can write <math>\lim_{x_0} f|_E</math>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>An amusing fact/potential advantage of the function restriction notation is that one can write it in a ''point-free notation''. Namely, we can write <math>\lim_{x_0} f|_E<ins style="font-weight: bold; text-decoration: none;"></math>. Actually, even with Tao's notation we could have written <math>\lim_{x_0;\, E} f</ins></math>.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>I think usually one would write the above like <math>\lim_{x\to 0+} |x|/x = 1</math>. So then one gets to keep anonymous functions, but at the expense of adding more complexity to the notation (one now needs to assign meaning to "<math>0+</math>").</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>I think usually one would write the above like <math>\lim_{x\to 0+} |x|/x = 1</math>. So then one gets to keep anonymous functions, but at the expense of adding more complexity to the notation (one now needs to assign meaning to "<math>0+</math>").</div></td></tr> </table>

IssaRice https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Tao%27s_notation_for_limits&diff=1016&oldid=prev IssaRice at 03:18, 1 December 2018 2018-12-01T03:18:27Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 03:18, 1 December 2018</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l12">Line 12:</td> <td colspan="2" class="diff-lineno">Line 12:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Rather than thinking of this as a result per se, I think it's better to think of this as ''alternative notations''. Why might one prefer one notation over the other? I think the strength of Tao's notation is that it works for anonymous functions/expressions. To be able to use the function restriction notation <math>f|_E</math>, we must have named our function beforehand. To give an example, we can write something like <math>\lim_{x\to 0;\, x\in (0,\infty)} |x|/x = 1</math>, but this is difficult to write in the other notation; we would have to say something like, "Let <math>f:\mathbf R\setminus \{0\} \to \mathbf R</math> be defined by <math>f(x) := |x|/x</math>. Then we have <math>\lim_{x\to 0} f|_{(0,\infty)}(x) = 1</math>."</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Rather than thinking of this as a result per se, I think it's better to think of this as ''alternative notations''. Why might one prefer one notation over the other? I think the strength of Tao's notation is that it works for anonymous functions/expressions. To be able to use the function restriction notation <math>f|_E</math>, we must have named our function beforehand. To give an example, we can write something like <math>\lim_{x\to 0;\, x\in (0,\infty)} |x|/x = 1</math>, but this is difficult to write in the other notation; we would have to say something like, "Let <math>f:\mathbf R\setminus \{0\} \to \mathbf R</math> be defined by <math>f(x) := |x|/x</math>. Then we have <math>\lim_{x\to 0} f|_{(0,\infty)}(x) = 1</math>."</div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">An amusing fact/potential advantage of the function restriction notation is that one can write it in a ''point-free notation''. Namely, we can write <math>\lim_{x_0} f|_E</math>.</ins></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>I think usually one would write the above like <math>\lim_{x\to 0+} |x|/x = 1</math>. So then one gets to keep anonymous functions, but at the expense of adding more complexity to the notation (one now needs to assign meaning to "<math>0+</math>").</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>I think usually one would write the above like <math>\lim_{x\to 0+} |x|/x = 1</math>. So then one gets to keep anonymous functions, but at the expense of adding more complexity to the notation (one now needs to assign meaning to "<math>0+</math>").</div></td></tr> </table>

IssaRice https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Tao%27s_notation_for_limits&diff=1015&oldid=prev IssaRice at 03:16, 1 December 2018 2018-12-01T03:16:38Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 03:16, 1 December 2018</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l14">Line 14:</td> <td colspan="2" class="diff-lineno">Line 14:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>I think usually one would write the above like <math>\lim_{x\to 0+} |x|/x = 1</math>. So then one gets to keep anonymous functions, but at the expense of adding more complexity to the notation (one now needs to assign meaning to "<math>0+</math>").</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>I think usually one would write the above like <math>\lim_{x\to 0+} |x|/x = 1</math>. So then one gets to keep anonymous functions, but at the expense of adding more complexity to the notation (one now needs to assign meaning to "<math>0+</math>").</div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">One should compare this to the function notation (<math>f'</math>) vs Leibniz notation (<math>\frac{d}{dx} f(x)</math>) for derivatives.</ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{| class="sortable wikitable"</ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">|-</ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">! Notation !! Symbols !! Strengths !! Weaknesses</ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">|-</ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">| Tao's notation || <math>\lim_{x\to x_0;\, x\in E} f(x)</math> || Works with anonymous functions. || Somewhat verbose: the "<math>x\in E</math>" part must be written out each time.</ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">|-</ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">| Function restriction notation || <math>\lim_{x\to x_0} f|_E(x)</math> || || Doesn't work with anonymous functions.</ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">|-</ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">| Standard notation || <math>\lim_{x\to x_0} f(x)</math> || Works with anonymous functions. ||</ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">|}</ins></div></td></tr> </table>

IssaRice https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Tao%27s_notation_for_limits&diff=1014&oldid=prev IssaRice at 03:11, 1 December 2018 2018-12-01T03:11:57Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 03:11, 1 December 2018</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l11">Line 11:</td> <td colspan="2" class="diff-lineno">Line 11:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Combining these two equalities, we have <math>\lim_{x \to x_0;\, x\in E} f(x) = \lim_{x\to x_0} f|_E(x)</math> as promised.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Combining these two equalities, we have <math>\lim_{x \to x_0;\, x\in E} f(x) = \lim_{x\to x_0} f|_E(x)</math> as promised.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Why might one prefer one notation over the other? I think the strength of Tao's notation is that it works for anonymous functions/expressions. To be able to use the function restriction notation <math>f|_E</math>, we must have named our function beforehand. To give an example, we can write something like <math>\lim_{x\to 0;\, x\in (0,\infty)} |x|/x = 1</math>, but this is difficult to write in the other notation; we would have to say something like, "Let <math>f:\mathbf R\setminus \{0\} \to \mathbf R</math> be defined by <math>f(x) := |x|/x</math>. Then we have <math>\lim_{x\to 0} f|_{(0,\infty)}(x) = 1</math>."</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Rather than thinking of this as a result per se, I think it's better to think of this as ''alternative notations''. </ins>Why might one prefer one notation over the other? I think the strength of Tao's notation is that it works for anonymous functions/expressions. To be able to use the function restriction notation <math>f|_E</math>, we must have named our function beforehand. To give an example, we can write something like <math>\lim_{x\to 0;\, x\in (0,\infty)} |x|/x = 1</math>, but this is difficult to write in the other notation; we would have to say something like, "Let <math>f:\mathbf R\setminus \{0\} \to \mathbf R</math> be defined by <math>f(x) := |x|/x</math>. Then we have <math>\lim_{x\to 0} f|_{(0,\infty)}(x) = 1</math>."</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>I think usually one would write the above like <math>\lim_{x\to 0+} |x|/x = 1</math>. So then one gets to keep anonymous functions, but at the expense of adding more complexity to the notation (one now needs to assign meaning to "<math>0+</math>").</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>I think usually one would write the above like <math>\lim_{x\to 0+} |x|/x = 1</math>. So then one gets to keep anonymous functions, but at the expense of adding more complexity to the notation (one now needs to assign meaning to "<math>0+</math>").</div></td></tr> </table>

IssaRice https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Tao%27s_notation_for_limits&diff=1013&oldid=prev IssaRice at 03:08, 1 December 2018 2018-12-01T03:08:36Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 03:08, 1 December 2018</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l13">Line 13:</td> <td colspan="2" class="diff-lineno">Line 13:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Why might one prefer one notation over the other? I think the strength of Tao's notation is that it works for anonymous functions/expressions. To be able to use the function restriction notation <math>f|_E</math>, we must have named our function beforehand. To give an example, we can write something like <math>\lim_{x\to 0;\, x\in (0,\infty)} |x|/x = 1</math>, but this is difficult to write in the other notation; we would have to say something like, "Let <math>f:\mathbf R\setminus \{0\} \to \mathbf R</math> be defined by <math>f(x) := |x|/x</math>. Then we have <math>\lim_{x\to 0} f|_{(0,\infty)}(x) = 1</math>."</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Why might one prefer one notation over the other? I think the strength of Tao's notation is that it works for anonymous functions/expressions. To be able to use the function restriction notation <math>f|_E</math>, we must have named our function beforehand. To give an example, we can write something like <math>\lim_{x\to 0;\, x\in (0,\infty)} |x|/x = 1</math>, but this is difficult to write in the other notation; we would have to say something like, "Let <math>f:\mathbf R\setminus \{0\} \to \mathbf R</math> be defined by <math>f(x) := |x|/x</math>. Then we have <math>\lim_{x\to 0} f|_{(0,\infty)}(x) = 1</math>."</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">On the other hand, </del>I think usually one would write the above like <math>\lim_{x\to 0+} |x|/x = 1</math>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>I think usually one would write the above like <math>\lim_{x\to 0+} |x|/x = 1</math><ins style="font-weight: bold; text-decoration: none;">. So then one gets to keep anonymous functions, but at the expense of adding more complexity to the notation (one now needs to assign meaning to "<math>0+</math>")</ins>.</div></td></tr> </table>

IssaRice https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Tao%27s_notation_for_limits&diff=1012&oldid=prev IssaRice at 03:07, 1 December 2018 2018-12-01T03:07:03Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 03:07, 1 December 2018</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l12">Line 12:</td> <td colspan="2" class="diff-lineno">Line 12:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Why might one prefer one notation over the other? I think the strength of Tao's notation is that it works for anonymous functions/expressions. To be able to use the function restriction notation <math>f|_E</math>, we must have named our function beforehand. To give an example, we can write something like <math>\lim_{x\to 0;\, x\in (0,\infty)} |x|/x = 1</math>, but this is difficult to write in the other notation; we would have to say something like, "Let <math>f:\mathbf R\setminus \{0\} \to \mathbf R</math> be defined by <math>f(x) := |x|/x</math>. Then we have <math>\lim_{x\to 0} f|_{(0,\infty)}(x) = 1</math>."</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Why might one prefer one notation over the other? I think the strength of Tao's notation is that it works for anonymous functions/expressions. To be able to use the function restriction notation <math>f|_E</math>, we must have named our function beforehand. To give an example, we can write something like <math>\lim_{x\to 0;\, x\in (0,\infty)} |x|/x = 1</math>, but this is difficult to write in the other notation; we would have to say something like, "Let <math>f:\mathbf R\setminus \{0\} \to \mathbf R</math> be defined by <math>f(x) := |x|/x</math>. Then we have <math>\lim_{x\to 0} f|_{(0,\infty)}(x) = 1</math>."</div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">On the other hand, I think usually one would write the above like <math>\lim_{x\to 0+} |x|/x = 1</math>.</ins></div></td></tr> </table>

IssaRice https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Tao%27s_notation_for_limits&diff=1011&oldid=prev IssaRice at 03:05, 1 December 2018 2018-12-01T03:05:27Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 03:05, 1 December 2018</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l11">Line 11:</td> <td colspan="2" class="diff-lineno">Line 11:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Combining these two equalities, we have <math>\lim_{x \to x_0;\, x\in E} f(x) = \lim_{x\to x_0} f|_E(x)</math> as promised.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Combining these two equalities, we have <math>\lim_{x \to x_0;\, x\in E} f(x) = \lim_{x\to x_0} f|_E(x)</math> as promised.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Why might one prefer one notation over the other? I think the strength of Tao's notation is that it works for anonymous functions/expressions. To be able to use the function restriction notation <math>f|_E</math>, we must have named our function beforehand. To give an example, we can write something like <math>\lim_{x\to 0;\, x\in (0,\infty)} |x|/x = 1</math>, but this is difficult to write in the other notation; we would have to say something like, "Let <math>f:\mathbf R\setminus \{0\} \to \mathbf R</math> be <del style="font-weight: bold; text-decoration: none;">define </del>by <math>f(x) := |x|/x</math>. Then we have <math>\lim_{x\to 0} f|_{(0,\infty)} = 1</math>."</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Why might one prefer one notation over the other? I think the strength of Tao's notation is that it works for anonymous functions/expressions. To be able to use the function restriction notation <math>f|_E</math>, we must have named our function beforehand. To give an example, we can write something like <math>\lim_{x\to 0;\, x\in (0,\infty)} |x|/x = 1</math>, but this is difficult to write in the other notation; we would have to say something like, "Let <math>f:\mathbf R\setminus \{0\} \to \mathbf R</math> be <ins style="font-weight: bold; text-decoration: none;">defined </ins>by <math>f(x) := |x|/x</math>. Then we have <math>\lim_{x\to 0} f|_{(0,\infty)}<ins style="font-weight: bold; text-decoration: none;">(x) </ins>= 1</math>."</div></td></tr> </table>

IssaRice https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Tao%27s_notation_for_limits&diff=1010&oldid=prev IssaRice at 03:04, 1 December 2018 2018-12-01T03:04:53Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 03:04, 1 December 2018</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l11">Line 11:</td> <td colspan="2" class="diff-lineno">Line 11:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Combining these two equalities, we have <math>\lim_{x \to x_0;\, x\in E} f(x) = \lim_{x\to x_0} f|_E(x)</math> as promised.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Combining these two equalities, we have <math>\lim_{x \to x_0;\, x\in E} f(x) = \lim_{x\to x_0} f|_E(x)</math> as promised.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Why might one prefer one notation over the other? I think the strength of Tao's notation is that it works for anonymous functions/expressions. To be able to use the function restriction notation <math>f|_E</math>, we must have named our function beforehand. To give an example, we can write something like <math>\lim_{x\to 0\, x\in (0,\infty)} |x|/x = 1</math>, but this is difficult to write in the other notation; we would have to say something like, "Let <math>f:\mathbf R\setminus \{0\} \to \mathbf R</math> be define by <math>f(x) := |x|/x</math>. Then we have <math>\lim_{x\to 0} f|_{(0,\infty)} = 1</math>."</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Why might one prefer one notation over the other? I think the strength of Tao's notation is that it works for anonymous functions/expressions. To be able to use the function restriction notation <math>f|_E</math>, we must have named our function beforehand. To give an example, we can write something like <math>\lim_{x\to 0<ins style="font-weight: bold; text-decoration: none;">;</ins>\, x\in (0,\infty)} |x|/x = 1</math>, but this is difficult to write in the other notation; we would have to say something like, "Let <math>f:\mathbf R\setminus \{0\} \to \mathbf R</math> be define by <math>f(x) := |x|/x</math>. Then we have <math>\lim_{x\to 0} f|_{(0,\infty)} = 1</math>."</div></td></tr> </table>

IssaRice