Math Problem Statement

Calculer la limite suivant et donner la résultat : lim⁡x→0(cos⁡(x)2−1)(cos⁡(x)3−1)⋅⋅(cos⁡(x)n−1)x2n−2\lim_{x\rarr0}\frac{\left(\sqrt[2]{\cos\left(x\right)}-1\right)\left(\sqrt[3]{\cos\left(x\right)}-1\right)\cdot\cdot\left(\sqrt[n]{\cos\left(x\right)}-1\right)}{x^{2n-2}}

Solution

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Math Problem Analysis

Mathematical Concepts

Calculus
Limits
Trigonometric Functions
Cosine Function
Series Expansions

Formulas

Limit of a function: \lim_{x\rightarrow a} f(x)
Series expansion of cos(x) near x = 0: \cos(x) \approx 1 - \frac{x^2}{2}
Root properties: \sqrt[n]{1 + u} \approx 1 + \frac{u}{n} (for small u)

Theorems

Taylor Series Expansion
L'Hopital's Rule
Binomial Approximation for small angles

Suitable Grade Level

University Level (Advanced Calculus)

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