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Compute the limit of the given function

Evaluate the limit.

    \[ \lim_{x \to 0^+} x^{\frac{e}{1 + \log x}}. \]


Using the definition and continuity of the exponential,

    \begin{align*}  \lim_{x \to 0^+} x^{\frac{e}{1 + \log x}} &= \lim_{x \to 0^+} e^{ \frac{e}{1+\log x} \log x } \\[9pt]  &= \exp \left( \lim_{x \to 0^+} \frac{e \log x}{1 + \log x} \right) \\[9pt]  &= \exp \left( \lim_{x \to 0^+} \frac{\frac{e}{x}}{\frac{1}{x}} \right) &(\text{L'Hopital}) \\[9pt]  &= \exp \left( \lim_{x \to 0^+} e \right) \\[9pt]  &= e^e. \end{align*}

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