设lim _(x arrow 0)(1+x+(f(x))/(x))^(1)/(x)=e^3，则lim _(x arrow 0)(1+(f(x))/(x))^(1)/(x)=______.
设$\lim _{x \rightarrow 0}\left(1+x+\frac{f(x)}{x}\right)^{\frac{1}{x}}=e^{3}$，则$\lim _{x \rightarrow 0}\left(1+\frac{f(x)}{x}\right)^{\frac{1}{x}}=\_\_\_\_\_\_$.
题目解答
答案
设 $ g(x) = x + \frac{f(x)}{x} $，由已知条件得
$\lim_{x \to 0} \left(1 + g(x)\right)^{\frac{1}{x}} = e^3$
取对数得
$\lim_{x \to 0} \frac{\ln(1 + g(x))}{x} = 3$
由泰勒展开 $\ln(1 + g(x)) \approx g(x)$（当 $g(x) \to 0$），得
$\lim_{x \to 0} \frac{g(x)}{x} = 3$
即
$\lim_{x \to 0} \left(1 + \frac{f(x)}{x^2}\right) = 3 \quad \Rightarrow \quad \lim_{x \to 0} \frac{f(x)}{x^2} = 2$
对于所求极限
$\lim_{x \to 0} \left(1 + \frac{f(x)}{x}\right)^{\frac{1}{x}}$
取对数得
$\lim_{x \to 0} \frac{\ln\left(1 + \frac{f(x)}{x}\right)}{x} = \lim_{x \to 0} \frac{\frac{f(x)}{x}}{x} = \lim_{x \to 0} \frac{f(x)}{x^2} = 2$
取指数得
$\lim_{x \to 0} \left(1 + \frac{f(x)}{x}\right)^{\frac{1}{x}} = e^2$
答案： $\boxed{e^2}$