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Rational limit

Let \mathbb{N} \ni n \geq 2. Evaluate the limit

    \[\ell = \lim_{x \rightarrow 1} \frac{\left ( 1-\sqrt{x} \right )\left ( 1-\sqrt[3]{x} \right )\left ( 1-\sqrt[4]{x} \right ) \cdots \left ( 1-\sqrt[n]{x} \right )}{\left ( 1-x \right )^{n-1}}\]

Solution

Let us consider the function f_n(x) = \sqrt[n]{x}. Then,

    \[\frac{1}{n} = f'_n(1) = \lim_{x\rightarrow 1} \frac{f_n(x) - f_n(1)}{x-1} = \lim_{x\rightarrow 1} \frac{1-\sqrt[n]{x}}{1-x}\]

Hence,

    \[\ell = f'_2(1) \cdot f'_3(1) \cdots f'_n(1) = \frac{1}{2} \cdot \frac{1}{3} \cdots \frac{1}{n} = \frac{1}{n!}\]

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