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"value": "Die Regel von L'Hospital ist eine Methode in der Mathematik, speziell in der Analysis, um Grenzwerte von Funktionen zu bestimmen, bei denen unbestimmte Ausdruecke wie 0/0 oder / auftreten. Die Regel ist nach dem franzoesischen Mathematiker Guillaume de l'Hospital benannt, der sie in seinem Buch Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes im Jahr 1696, dem ersten Lehrbuch der Differentialrechnung, veroeffentlichte. \n\nDie Regel besagt, dass der Grenzwert eines Quotienten zweier Funktionen, die beide gegen Null oder beide gegen Unendlich streben, unter bestimmten Bedingungen gleich dem Grenzwert des Quotienten ihrer Ableitungen ist. Mathematisch formuliert heisst das:\n\nWenn \\( \\lim_{x \\to a} f(x) = \\lim_{x \\to a} g(x) = 0 \\) oder \\( \\lim_{x \\to a} f(x) = \\lim_{x \\to a} g(x) = \\infty \\) und die Ableitungen \\( f' \\) und \\( g' \\) existieren nahe \\( a \\) (und \\( g'(x) \\neq 0 \\)), dann gilt:\n\\[ \\lim_{x \\to a} \\frac{f(x)}{g(x)} = \\lim_{x \\to a} \\frac{f'(x)}{g'(x)}, \\]\nsofern der rechts stehende Grenzwert existiert oder gegen unendlich strebt.\n\nDiese Regel ist besonders nuetzlich in der hoeheren Mathematik und bei der Loesung komplexerer Grenzwertprobleme, wo direkte Berechnungen oft schwierig oder unmoeglich sind."
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