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求 $lim_{x \to 0} (1 + x e^{x})^{\frac{x + 1}{x}}$ .

Posted by haifeng on 2019-10-27 10:43:43 last update 2019-10-27 10:44:00 | Edit | Answers (1)

求 $lim_{x \to 0} (1 + x e^{x})^{\frac{x + 1}{x}}$ .

Posted by haifeng on 2019-10-27 10:50:08

当 $x \to 0$ 时, $x e^{x} \to 0$ ,

$lim_{x \to 0} (1 + x e^{x})^{\frac{x + 1}{x}} = lim_{x \to 0} [(1 + x e^{x})^{\frac{1}{x e^{x}}}]^{e^{x} (x + 1)} = e^{1} = e .$