题目
(10)求lim_(xtoinfty)x^2[e^(1+(1)/(x))^(x)-(1+(1)/(x))^ex].
(10)求$\lim_{x\to\infty}x^{2}[e^{(1+\frac{1}{x})^{x}}-(1+\frac{1}{x})^{ex}]$.
题目解答
答案
令 $y = \frac{1}{x}$，则 $x \to \infty$ 时 $y \to 0$。考虑表达式：
\[
\lim_{y \to 0} \frac{e^{\left(1 + y\right)^{\frac{1}{y}}} - \left(1 + y\right)^{\frac{e}{y}}}{y^2}.
\]
利用泰勒展开：
\[
\left(1 + y\right)^{\frac{1}{y}} = e \left(1 - \frac{y}{2} + O(y^2)\right), \quad \left(1 + y\right)^{\frac{e}{y}} = e^e \left(1 - \frac{ey}{2} + O(y^2)\right).
\]
进一步展开指数项：
\[
e^{\left(1 + y\right)^{\frac{1}{y}}} = e^e \left(1 - \frac{ey}{2} + \frac{e^2y^2}{8} + O(y^2)\right).
\]
两式相减：
\[
e^{\left(1 + y\right)^{\frac{1}{y}}} - \left(1 + y\right)^{\frac{e}{y}} = e^e \left(\frac{e^2y^2}{8} + O(y^2)\right).
\]
除以 $y^2$ 并取极限：
\[
\lim_{y \to 0} \frac{e^e \left(\frac{e^2y^2}{8} + O(y^2)\right)}{y^2} = \frac{e^{e+2}}{8}.
\]
**答案：** $\boxed{\frac{e^{e+2}}{8}}$
解析
步骤 1：变量替换
令 $y = \frac{1}{x}$，则 $x \to \infty$ 时 $y \to 0$。原极限问题转化为求 \[ \lim_{y \to 0} \frac{e^{\left(1 + y\right)^{\frac{1}{y}}} - \left(1 + y\right)^{\frac{e}{y}}}{y^2}. \]
步骤 2：泰勒展开
利用泰勒展开，我们有 \[ \left(1 + y\right)^{\frac{1}{y}} = e \left(1 - \frac{y}{2} + O(y^2)\right), \quad \left(1 + y\right)^{\frac{e}{y}} = e^e \left(1 - \frac{ey}{2} + O(y^2)\right). \]
步骤 3：指数项展开
进一步展开指数项，得到 \[ e^{\left(1 + y\right)^{\frac{1}{y}}} = e^e \left(1 - \frac{ey}{2} + \frac{e^2y^2}{8} + O(y^2)\right). \]
步骤 4：相减并取极限
两式相减，得到 \[ e^{\left(1 + y\right)^{\frac{1}{y}}} - \left(1 + y\right)^{\frac{e}{y}} = e^e \left(\frac{e^2y^2}{8} + O(y^2)\right). \] 除以 $y^2$ 并取极限，得到 \[ \lim_{y \to 0} \frac{e^e \left(\frac{e^2y^2}{8} + O(y^2)\right)}{y^2} = \frac{e^{e+2}}{8}. \]
令 $y = \frac{1}{x}$，则 $x \to \infty$ 时 $y \to 0$。原极限问题转化为求 \[ \lim_{y \to 0} \frac{e^{\left(1 + y\right)^{\frac{1}{y}}} - \left(1 + y\right)^{\frac{e}{y}}}{y^2}. \]
步骤 2：泰勒展开
利用泰勒展开，我们有 \[ \left(1 + y\right)^{\frac{1}{y}} = e \left(1 - \frac{y}{2} + O(y^2)\right), \quad \left(1 + y\right)^{\frac{e}{y}} = e^e \left(1 - \frac{ey}{2} + O(y^2)\right). \]
步骤 3：指数项展开
进一步展开指数项，得到 \[ e^{\left(1 + y\right)^{\frac{1}{y}}} = e^e \left(1 - \frac{ey}{2} + \frac{e^2y^2}{8} + O(y^2)\right). \]
步骤 4：相减并取极限
两式相减，得到 \[ e^{\left(1 + y\right)^{\frac{1}{y}}} - \left(1 + y\right)^{\frac{e}{y}} = e^e \left(\frac{e^2y^2}{8} + O(y^2)\right). \] 除以 $y^2$ 并取极限，得到 \[ \lim_{y \to 0} \frac{e^e \left(\frac{e^2y^2}{8} + O(y^2)\right)}{y^2} = \frac{e^{e+2}}{8}. \]