QCM corrigé Analyse numérique licence master 0 analyse numérique, arithmétique, continuité, denombrement, dénombrement, mathematique, nombres complexes, probabilité, QCM licence math, QCM Master math, series, suites 06 h 58 <html xml:lang="en" xmlns="http://www.w3.org/1998/Math/MathML"> <head> <title>QCM Analyse numérique licence master </title> <script src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_SVG" type="text/javascript"></script> </head> <body> <!-- MathMagic Pro Win v8.02, MathML 2.0 converter, 2015.8.21 14:18 --> <math xmlns="http://www.w3.org/1998/Math/MathML"> <semantics> <mstyle> <mtable> <mtr> <mtd columnalign="left"> <mi mathcolor="#0000ff">Q</mi> <mi mathcolor="#0000ff">u</mi> <mi mathcolor="#0000ff">e</mi> <mi mathcolor="#0000ff">s</mi> <mi mathcolor="#0000ff">t</mi> <mi mathcolor="#0000ff">i</mi> <mi mathcolor="#0000ff">o</mi> <mi mathcolor="#0000ff">n</mi> <mn mathcolor="#0000ff">1</mn> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mi>On</mi> 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mathvariant="italic">α</mo> <mo>></mo> <mn>2</mn> <mo>/</mo> <mn>3</mn> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mo>□</mo> <mo mathvariant="italic">α</mo> <mo>></mo> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mi mathcolor="#0000ff">Question</mi> <mspace width="0.33em"> <mn mathcolor="#0000ff">8</mn> </mspace></mtd> </mtr> <mtr> <mtd columnalign="left"> <mi>Soit</mi> <mspace width="0.33em"> <mi>l</mi> <mi>'</mi> <mi>a</mi> <mi>p</mi> <mi>p</mi> <mi>l</mi> <mi>i</mi> <mi>c</mi> <mi>a</mi> <mi>t</mi> <mi>i</mi> <mi>o</mi> <mi>n</mi> <mspace width="0.33em"> <mi>f</mi> <mspace width="0.33em"> <mi>dans</mi> <mspace width="0.33em"> <mo>ℂ</mo> <mspace width="0.33em"> <mspace width="0.33em"> <mi>définie</mi> <mspace width="0.33em"> <mi>par</mi> <mspace width="0.33em"> <mi>f</mi> <mi>(</mi> <mi>z</mi> <mi>)</mi> <mo>=</mo> <mstyle mathsize="14.0pt"> <mfrac> <mrow> <mi>z</mi> <mo>+</mo> <mi>i</mi> </mrow> <mrow> <mi>z</mi> <mo>−</mo> <mi>i</mi> </mrow> </mfrac> </mstyle> <mspace width="0.33em"> <mi>pour</mi> <mspace width="0.33em"> <mi>tout</mi> <mspace width="0.33em"> <mi>z</mi> <mspace width="0.33em"> <mi>different</mi> <mspace width="0.33em"> <mi>de</mi> <mspace width="0.33em"> <mi>i</mi> </mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mtd> </mtr> <mtr> <mtd columnalign="left"> <mi>et</mi> <mspace width="0.33em"> <mi>on</mi> <mspace width="0.33em"> <mi>considére</mi> <mspace width="0.33em"> <mi>E</mi> <mo>=</mo> <mfenced close="" open="{"> <mrow> <mi>z</mi> <mo>∊</mo> </mrow> </mfenced> <mo>ℂ</mo> <mspace width="0.33em"> <mi>,</mi> <mo>∣</mo> <mi>z</mi> <mo>∣</mo> <mo>=</mo> <mn>1</mn> <mfenced close="}" open=""> <mspace> </mspace></mfenced> <mspace width="0.33em"> <mi>et</mi> <mspace width="0.33em"> <mi>i</mi> <mo>ℝ</mo> <mo>=</mo> <mfenced close="" open="{"> <mrow> <mi>z</mi> <mo>∊</mo> </mrow> </mfenced> <mo>ℂ</mo> <mspace width="0.33em"> <mi>,</mi> <mi>Rel</mi> <mfenced close=")" open="("> <mi>z</mi> </mfenced> <mo>=</mo> <mn>0</mn> <mfenced close="}" open=""> <mspace> </mspace></mfenced> </mspace></mspace></mspace></mspace></mspace></mspace></mspace></mtd> </mtr> <mtr> <mtd columnalign="left"> <mo>□</mo> <mi>f</mi> <mspace width="0.33em"> <mi>réalise</mi> <mspace width="0.33em"> <mi>une</mi> <mspace width="0.33em"> <mi>bijection</mi> <mspace width="0.33em"> <mi>de</mi> <mspace width="0.33em"> <mo>ℂ</mo> <mo>∖</mo> <mfenced close="" open="{"> <mi>i</mi> </mfenced> <mfenced close="}" open=""> <mspace> </mspace></mfenced> <mspace width="0.33em"> <mi>dans</mi> <mspace width="0.33em"> <mo>ℂ</mo> <mi>.</mi> </mspace></mspace></mspace></mspace></mspace></mspace></mspace></mtd> </mtr> <mtr> <mtd columnalign="left"> <mo>□</mo> <mi>f</mi> <mfenced close=")" open="("> <mrow> <mi>i</mi> <mo>ℝ</mo> </mrow> </mfenced> <mo>=</mo> <mo>ℝ</mo> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mo>□</mo> <mi>f</mi> <mfenced close=")" open="("> <mrow> <mi>i</mi> <mo>ℝ</mo> </mrow> </mfenced> <mo>=</mo> <mi>E</mi> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mo>□</mo> <mi>f</mi> <mfenced close=")" open="("> <mi>E</mi> </mfenced> <mo>=</mo> <mi>i</mi> <mo>ℝ</mo> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mo>□</mo> <mi>Aucune</mi> <mspace width="0.33em"> <mi>réponses</mi> <mspace width="0.33em"> <mi>n</mi> <mi>'</mi> <mi>e</mi> <mi>s</mi> <mi>t</mi> <mspace width="0.33em"> <mi>c</mi> <mi>o</mi> <mi>r</mi> <mi>r</mi> <mi>e</mi> <mi>c</mi> <mi>t</mi> <mi>e</mi> <mi>.</mi> </mspace></mspace></mspace></mtd> </mtr> <mtr> <mtd columnalign="left"> <mi mathcolor="#0000ff">Question</mi> <mspace width="0.33em"> <mn mathcolor="#0000ff">9</mn> </mspace></mtd> </mtr> <mtr> <mtd columnalign="left"> <mi>On</mi> <mspace width="0.33em"> <mi>désigne</mi> <mspace width="0.33em"> <mi>P</mi> <mfenced close=")" open="("> <mi>E</mi> </mfenced> <mspace width="0.33em"> <mi>l</mi> <mi>'</mi> <mi>e</mi> <mi>n</mi> <mi>s</mi> <mi>e</mi> <mi>m</mi> <mi>b</mi> <mi>l</mi> <mi>e</mi> <mspace width="0.33em"> <mi>des</mi> <mspace width="0.33em"> <mi>parties</mi> <mspace width="0.33em"> <mi>de</mi> <mspace width="0.33em"> <mi>E</mi> <mi>.</mi> </mspace></mspace></mspace></mspace></mspace></mspace></mspace></mtd> </mtr> <mtr> <mtd columnalign="left"> <mi>Soit</mi> <mspace width="0.33em"> <mi>E</mi> <mspace width="0.33em"> <mi>une</mi> <mspace width="0.33em"> <mi>ensemble</mi> <mspace width="0.33em"> <mi>de</mi> <mspace width="0.33em"> <mi>cardinal</mi> <mspace width="0.33em"> <mi>fini</mi> <mspace width="0.33em"> <mi>n</mi> <mi>.</mi> </mspace></mspace></mspace></mspace></mspace></mspace></mspace></mtd> </mtr> <mtr> <mtd columnalign="left"> <mi>le</mi> <mspace width="0.33em"> <mi>nombre</mi> <mspace width="0.33em"> <mi>de</mi> <mspace width="0.33em"> <mi>couples</mi> <mspace width="0.33em"> <mfenced close=")" open="("> <mrow> <mi>A</mi> <mi>,</mi> <mi>B</mi> </mrow> </mfenced> <mo mathsize="14.0pt">∊</mo> <mi>P</mi> <mfenced close=")" open="("> <mi>E</mi> </mfenced> <mo>×</mo> <mi>P</mi> <mfenced close=")" open="("> <mi>E</mi> </mfenced> <mspace width="0.33em"> <mi>tel</mi> <mspace width="0.33em"> <mi>que</mi> <mspace width="0.33em"> <mi>A</mi> <mo>⊂</mo> <mi>B</mi> <mspace width="0.33em"> <mi>v</mi> <mi>a</mi> <mi>u</mi> <mi>t</mi> <mi>:</mi> </mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mtd> </mtr> <mtr> <mtd columnalign="left"> <mo>□</mo> <msup> <mn>2</mn> <mi>n</mi> </msup> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mo>□</mo> <msup> <mn>3</mn> <mrow> <mi>n</mi> <mo mathsize="50.0%">−</mo> <mn>1</mn> </mrow> </msup> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mo>□</mo> <msup> <mn>3</mn> <mi>n</mi> </msup> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mo>□</mo> <msup> <mn>2</mn> <msup> <mi>n</mi> <mn>2</mn> </msup> </msup> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mo>□</mo> <msup> <mrow> <mi>n</mi> <mn>2</mn> </mrow> <mrow> <mi>n</mi> <mo mathsize="50.0%">−</mo> <mn>1</mn> </mrow> </msup> </mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mtd> </mtr> <mtr> <mtd columnalign="left"> <mspace> </mspace></mtd> </mtr> <mtr> <mtd columnalign="left"> <mi mathcolor="#0000ff">Question</mi> <mspace width="0.33em"> <mn mathcolor="#0000ff">10</mn> </mspace></mtd> </mtr> <mtr> <mtd columnalign="left"> <mi>Une</mi> <mspace width="0.33em"> <mi>urne</mi> <mspace width="0.33em"> <mi>contient</mi> <mspace width="0.33em"> <mn>2</mn> <mspace width="0.33em"> <mi>boules</mi> <mspace width="0.33em"> <mi>blanches</mi> <mspace width="0.33em"> <mi>et</mi> <mspace width="0.33em"> <mfenced close=")" open="("> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfenced> <mspace width="0.33em"> <mi>boules</mi> <mspace width="0.33em"> <mi>r</mi> <mi>o</mi> <mi>u</mi> <mi>g</mi> <mi>e</mi> <mi>s</mi> <mi>.</mi> </mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mtd> </mtr> <mtr> <mtd columnalign="left"> <mi>On</mi> <mspace width="0.33em"> <mi>effectue</mi> <mspace width="0.33em"> <mi>des</mi> <mspace width="0.33em"> <mi>tirages</mi> <mspace width="0.33em"> <mi>successifs</mi> <mspace width="0.33em"> <mi>sans</mi> <mspace width="0.33em"> <mi>remise</mi> <mspace width="0.33em"> <mi>d</mi> <mi>'</mi> <mi>u</mi> <mi>n</mi> <mi>e</mi> <mspace width="0.33em"> <mi>boule</mi> <mspace width="0.33em"> <mi>et</mi> <mspace width="0.33em"> <mi>on</mi> <mspace width="0.33em"> <mi>note</mi> <mspace width="0.33em"> <mi>X</mi> <mspace width="0.33em"> </mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mtd> </mtr> <mtr> <mtd columnalign="left"> <mi>le</mi> <mspace width="0.33em"> <mi>rang</mi> <mspace width="0.33em"> <mi>pour</mi> <mspace width="0.33em"> <mi>lequel</mi> <mspace width="0.33em"> <mi>la</mi> <mspace width="0.33em"> <mi>boule</mi> <mspace width="0.33em"> <mi>blanche</mi> <mspace width="0.33em"> <mi>apparait</mi> <mspace width="0.33em"> <mi>pour</mi> <mspace width="0.33em"> <mi>la</mi> <mspace width="0.33em"> <mi>premiere</mi> <mspace width="0.33em"> <mi>f</mi> <mi>o</mi> <mi>i</mi> <mi>s</mi> <mi>.</mi> </mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mtd> </mtr> <mtr> <mtd columnalign="left"> <mi>E</mi> <mfenced close=")" open="("> <mi>X</mi> </mfenced> <mo>=</mo> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mo>□</mo> <mstyle mathsize="14.0pt"> <mfrac> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mstyle> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mo>□</mo> <mstyle mathsize="14.0pt"> <mfrac> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mstyle> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mo>□</mo> <mstyle mathsize="14.0pt"> <mfrac> <mrow> <mn>3</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mstyle> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mo>□</mo> <mstyle mathsize="14.0pt"> <mfrac> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>3</mn> </mfrac> </mstyle> </mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mtd> </mtr> <mtr> <mtd columnalign="left"> <mi mathcolor="#0000ff">Question</mi> <mspace width="0.33em"> <mn mathcolor="#0000ff">11</mn> </mspace></mtd> </mtr> <mtr> <mtd columnalign="left"> <mi>Soit</mi> <mspace width="0.33em"> <mi>S</mi> <mspace width="0.33em"> <mi>l</mi> <mi>'</mi> <mi>ensemble</mi> <mspace width="0.33em"> <mi>des</mi> <mspace width="0.33em"> <mi>entiers</mi> <mspace width="0.33em"> <mi>x</mi> <mspace width="0.33em"> <mi>tel</mi> <mspace width="0.33em"> <mi>que</mi> <mspace width="0.33em"> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>−</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mspace width="0.33em"> <mi>est</mi> <mspace width="0.33em"> <mi>divisible</mi> <mspace width="0.33em"> <mi>par</mi> <mspace width="0.33em"> <mn>7</mn> <mi>.</mi> </mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mtd> </mtr> <mtr> <mtd columnalign="left"> <mi>S</mi> <mspace width="0.33em"> <mi>est</mi> <mspace width="0.33em"> <mi>alors</mi> <mspace width="0.33em"> <mi>:</mi> </mspace></mspace></mspace></mtd> </mtr> <mtr> <mtd columnalign="left"> <mo>□</mo> <mi>S</mi> <mo>=</mo> <mfenced close="}" open="{"> <mrow> <mn>7</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mi>,</mi> <mn>7</mn> <mi>k</mi> <mo>+</mo> <mn>5</mn> </mrow> </mfenced> <mi>,</mi> <mi>k</mi> <mo>∊</mo> <mo>ℤ</mo> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mo>□</mo> <mi>S</mi> <mo>=</mo> <mfenced close="}" open="{"> <mrow> <mn>7</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mi>,</mi> <mn>7</mn> <mi>k</mi> <mo>+</mo> <mn>5</mn> </mrow> </mfenced> <mi>,</mi> <mi>k</mi> <mo>∊</mo> <mo>ℤ</mo> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mo>□</mo> <mi>S</mi> <mo>=</mo> <mfenced close="}" open="{"> <mrow> <mn>7</mn> <mi>k</mi> <mo>+</mo> <mn>3</mn> <mi>,</mi> <mn>7</mn> <mi>k</mi> <mo>+</mo> <mn>5</mn> </mrow> </mfenced> <mi>,</mi> <mi>k</mi> <mo>∊</mo> <mo>ℤ</mo> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mo>□</mo> <mi>S</mi> <mo>=</mo> <mfenced close="}" open="{"> <mrow> <mn>7</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mi>,</mi> <mn>7</mn> <mi>k</mi> <mo>+</mo> <mn>3</mn> </mrow> </mfenced> <mi>,</mi> <mi>k</mi> <mo>∊</mo> <mo>ℤ</mo> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mi mathcolor="#0000ff">Question</mi> <mspace width="0.33em"> <mn mathcolor="#0000ff">12</mn> </mspace></mtd> </mtr> <mtr> <mtd columnalign="left"> <munder> <mi>lim</mi> <mrow> <mi>n</mi> <mo mathsize="50.0%">→</mo> <mo mathsize="50.0%">+</mo> <mo mathsize="50.0%">∞</mo> </mrow> </munder> <mspace width="0.33em"> <mrow> <munderover> <mo>∑</mo> <mrow> <mi>k</mi> <mo mathsize="8.0pt">=</mo> <mn>0</mn> </mrow> <mi>n</mi> </munderover> <mspace> </mspace></mrow> <mi>sin</mi> <mfenced close=")" open="("> <mstyle mathsize="14.0pt"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> </mfrac> </mstyle> </mfenced> <mo>=</mo> </mspace></mtd> </mtr> <mtr> <mtd columnalign="left"> <mspace> </mspace></mtd> </mtr> <mtr> <mtd columnalign="left"> <mo>□</mo> <mn>0</mn> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mo>□</mo> <mi>sin</mi> <mfenced close=")" open="("> <mn>1</mn> </mfenced> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mo>□</mo> <mi>ln</mi> <mfenced close=")" open="("> <mn>2</mn> </mfenced> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mo>□</mo> <mn>1</mn> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mspace width="0.33em"> <mo>□</mo> <mo>+</mo> <mo>∞</mo> </mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mtd> </mtr> <mtr> <mtd columnalign="left"> <mi mathcolor="#0000ff">Question</mi> <mspace width="0.33em"> <mn mathcolor="#0000ff">13</mn> </mspace></mtd> </mtr> <mtr> <mtd columnalign="left"> <mi>Soit</mi> <mspace width="0.33em"> <mi>une</mi> <mspace width="0.33em"> <mi>fonction</mi> <mspace width="0.33em"> <mi>f</mi> <mspace width="0.33em"> <mi>d</mi> <mo>é</mo> <mi>rivable</mi> <mspace width="0.33em"> <mi>en</mi> <mspace width="0.33em"> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0</mn> <mspace width="0.33em"> <mi>d</mi> <mo>é</mo> <mi>finie</mi> <mspace width="0.33em"> <mi>de</mi> <mspace width="0.33em"> <mo>ℝ</mo> <mo>→</mo> <mo>ℝ</mo> <mspace width="0.33em"> <mi>tel</mi> <mspace width="0.33em"> <mi>que</mi> <mspace width="0.33em"> <mi>f</mi> <mfenced close=")" open="("> <mn>0</mn> </mfenced> <mo>=</mo> <mn>0</mn> <mi>et</mi> <mspace width="0.33em"> <mi>f</mi> <mi>'(</mi> <mn>0</mn> <mi>)</mi> <mo>=</mo> <mn>1</mn> <mspace width="0.33em"> </mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mspace></mtd> </mtr> <mtr> <mtd columnalign="left"> <mspace> </mspace></mtd> </mtr> <mtr> <mtd columnalign="left"> <mo>∀</mo> <mi>n</mi> <mo>∊</mo> <msup> <mo>ℕ</mo> <mo>*</mo> </msup> <mspace width="0.33em"> <munder> <mi>lim</mi> <mrow> <mi mathsize="8.0pt">x</mi> <mo mathsize="8.0pt">→</mo> <mn mathsize="8.0pt">0</mn> </mrow> </munder> <mspace width="0.33em"> <mfrac> <mrow> <mi 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