已知函数 $f(x, y)=\ln (y+|x \sin y|)$, 则
$ \text{A.} $ $\left.\frac{\partial f}{\partial x}\right|_{(0,1)}$ 不存在, $\left.\frac{\partial f}{\partial y}\right|_{(0,1)}$ 存在 $ \text{B.} $ $\left.\frac{\partial f}{\partial x}\right|_{(0,1)}$ 存在, $\left.\frac{\partial f}{\partial y}\right|_{(0,1)}$ 不存在 $ \text{C.} $ $\left.\frac{\partial f}{\partial x}\right|_{(0,1)},\left.\frac{\partial f}{\partial y}\right|_{(0.1)}$ 均存在 $ \text{D.} $ $\left.\frac{\partial f}{\partial x}\right|_{(0.1)},\left.\frac{\partial f}{\partial y}\right|_{(0.1)}$ 均不存在
【答案】 A

【解析】 $f(0,1)=0$, 由偏导数的定义
$$
\left.\frac{\partial f}{\partial x}\right|_{(0,1)}=\lim _{x \rightarrow 0} \frac{f(x, 1)-f(0,1)}{x}=\lim _{x \rightarrow 0} \frac{\ln (1+\sin 1|x|)}{x}=\sin 1 \lim _{x \rightarrow 0} \frac{|x|}{x},
$$
因为 $\lim _{x \rightarrow 0^*} \frac{|x|}{x}=1, \lim _{x \rightarrow 0} \frac{|x|}{x}=-1$, 所以 $\left.\frac{\partial f}{\partial x}\right|_{(0.1)}$ 不存在, $\left.\frac{\partial f}{\partial y}\right|_{(0.1)}=\lim _{y \rightarrow 1} \frac{f(0, y)-f(0,1)}{y-1}=\lim _{y \rightarrow 1} \frac{\ln y}{y-1}=\lim _{y \rightarrow 1} \frac{y-1}{y-1}=1$, 所以 $\left.\frac{\partial f}{\partial y}\right|_{(0,1)}$ 存在.
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