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A growth model contains the factor xˣ for small positive x…

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A growth model contains the factor $x^x$ for small positive $x$ . At $x = 0$ the factor has the form $0^0$ , which is indeterminate. A student writes $x^x = e^{x\ln x}$ and studies the exponent $x\ln x$ , which has the form $0 \cdot \infty$ as $x \to 0^+$ . Find the limit of $x\ln x$ as $x \to 0^+$ . Then find the limit of $x^x$ as $x \to 0^+$ . Finally find the value of $x^x$ at $x = 0.01$ , to three decimal places.

Need a reminder? The key ideaShow

Series for small $x$ : $\sin x = x - \dfrac{x^3}{6} + \dots$ , $e^x = 1 + x + \dfrac{x^2}{2} + \dfrac{x^3}{6} + \dots$ . L'Hôpital's rule: for $\dfrac{0}{0}$ or $\dfrac{\infty}{\infty}$ , $\lim\dfrac{f}{g} = \lim\dfrac{f'}{g'}$ if the second limit exists.

The worked solution

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