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S:L15.231 - Revision history
2026-05-01T06:38:34Z
Revision history for this page on the wiki
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https://wiki.universitas.ro/index.php?title=S:L15.231&diff=9161&oldid=prev
Andrei.Horvat: final
2024-01-07T10:59:39Z
<p>final</p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 10:59, 7 January 2024</td>
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<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>''Fie <math>\left(a_n\right)_{n\ge 1}</math> un șir crescător de numere reale strict pozitive cu <math>\lim\limits_{n\to \infty} a_n = a</math>. Arătați că <math display="block">\lim\limits_{n\to\infty} \dfrac{a-a_n}{\ln\frac{a}{a_n}} = a</math>'''''Soluție'''</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>''Fie <math>\left(a_n\right)_{n\ge 1}</math> un șir crescător de numere reale strict pozitive cu <math>\lim\limits_{n\to \infty} a_n = a</math>. Arătați că <math display="block">\lim\limits_{n\to\infty} \dfrac{a-a_n}{\ln\frac{a}{a_n}} = a</math>'''''Soluție'''</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;">Fie <math>n \in \mathbb{N}^\ast</math>. Se consideră funcția <math>f_n:\left[a_n,a\right] \to \mathbb{R}</math>, cu <math>f_n\left(x\right) = \ln x</math>. Pentru această funcție se aplică [https://mathworld.wolfram.com/Mean-ValueTheorem.html teorema lui Lagrange], deci există <math>c_n \in \left(a_n, a\right)</math> astfel încât <math display="block">\frac{\ln a - \ln a_n}{a-a_n} = \frac{1}{c_n}.</math> Cum <math>\lim\limits_{n\to \infty} c_n = a</math> se obține <math display="block"> \lim\limits_{n\to\infty} \dfrac{a-a_n}{\ln\frac{a}{a_n}} = \lim\limits_{n\to \infty} c_n = a.</math></ins></div></td></tr>
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Andrei.Horvat
https://wiki.universitas.ro/index.php?title=S:L15.231&diff=9160&oldid=prev
Andrei.Horvat at 10:51, 7 January 2024
2024-01-07T10:51:28Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 10:51, 7 January 2024</td>
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<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>'''S:L15.231 (Andrei Horvat-Marc)'''</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>'''S:L15.231 (Andrei Horvat-Marc)'''</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>''Fie <math>\left(a_n\right)_{n\ge 1}</math> un șir crescător de numere reale strict pozitive cu <math>\lim\limits_{n\to \infty} a_n = a</math>. Arătați că <math display="block">\lim\limits_{n\to\infty} \dfrac{a-a_n}{\ln\frac{a}{a_n}} = a</math><del style="font-weight: bold; text-decoration: none;">.</del>''</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>''Fie <math>\left(a_n\right)_{n\ge 1}</math> un șir crescător de numere reale strict pozitive cu <math>\lim\limits_{n\to \infty} a_n = a</math>. Arătați că <math display="block">\lim\limits_{n\to\infty} \dfrac{a-a_n}{\ln\frac{a}{a_n}} = a</math><ins style="font-weight: bold; text-decoration: none;">'''''Soluție'</ins>''</div></td></tr>
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Andrei.Horvat
https://wiki.universitas.ro/index.php?title=S:L15.231&diff=9159&oldid=prev
Andrei.Horvat at 10:50, 7 January 2024
2024-01-07T10:50:31Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 10:50, 7 January 2024</td>
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<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>''Fie <math>\left(a_n\right)_{n\ge 1}</math> un șir crescător de numere reale strict pozitive cu <math>\lim\limits_{n\to \infty} a_n = a</math>. Arătați că <del style="font-weight: bold; text-decoration: none;"><nowiki></del><math>\lim\limits_{n\to\infty} \dfrac{a-a_n}{\ln\frac{a}{a_n}} = a<math>.<del style="font-weight: bold; text-decoration: none;"></nowiki></del>''</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>''Fie <math>\left(a_n\right)_{n\ge 1}</math> un șir crescător de numere reale strict pozitive cu <math>\lim\limits_{n\to \infty} a_n = a</math>. Arătați că <math <ins style="font-weight: bold; text-decoration: none;">display="block"</ins>>\lim\limits_{n\to\infty} \dfrac{a-a_n}{\ln\frac{a}{a_n}} = a<<ins style="font-weight: bold; text-decoration: none;">/</ins>math>.''</div></td></tr>
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Andrei.Horvat
https://wiki.universitas.ro/index.php?title=S:L15.231&diff=9158&oldid=prev
Andrei.Horvat at 10:49, 7 January 2024
2024-01-07T10:49:32Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 10:49, 7 January 2024</td>
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<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;">''Fie <math>\left(a_n\right)_{n\ge 1}</math> un șir crescător de numere reale strict pozitive cu <math>\lim\limits_{n\to \infty} a_n = a</math>. Arătați că <nowiki><math>\lim\limits_{n\to\infty} \dfrac{a-a_n}{\ln\frac{a}{a_n}} = a<math>.</nowiki>''</ins></div></td></tr>
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Andrei.Horvat
https://wiki.universitas.ro/index.php?title=S:L15.231&diff=9157&oldid=prev
Andrei.Horvat: Pagină nouă: '''S:L15.231 (Andrei Horvat-Marc)'''
2024-01-07T10:47:15Z
<p>Pagină nouă: '''S:L15.231 (Andrei Horvat-Marc)'''</p>
<p><b>New page</b></p><div>'''S:L15.231 (Andrei Horvat-Marc)'''</div>
Andrei.Horvat