题目
下列计算正确的是(). A. lim _(x arrow 0) 3 x^2 cdot ln x=lim _(x arrow 0) 3 cdot (ln x)/(x^2)=lim _(x arrow 0)-(3)/(2) cdot (x^-1)/(x^-3)=0.B. lim _(x arrow 0^+) (ln x)/(ln sin x)=lim _(x arrow 0^+) (x^-1)/(cot x)=lim _(x arrow 0^+) (tan x)/(x)=1.C. lim _(x arrow 0)((1)/(x)-(1)/(e^x)-1)=lim _(x arrow 0) (e^x-1-x)/(x e^x)-x=lim _(x arrow 0) (e^x-1)/(e^x)-1+x e^(x)=lim _(x arrow 0) (e^x)/(2 e^x)+x e^(x)=(1)/(2).D. lim _(x arrow 1) (x^3-3 x+2)/(x^3)-x^(2-x+1)=lim _(x arrow 1) (3 x^2-3)/(3 x^2)-2 x-1=lim _(x arrow 1) (6 x)/(6 x-2)=lim _(x arrow 1) (6)/(6)=1.
下列计算正确的是().
- A. $\lim _{x \rightarrow 0} 3 x^{2} \cdot \ln x=\lim _{x \rightarrow 0} 3 \cdot \frac{\ln x}{x^{2}}=\lim _{x \rightarrow 0}-\frac{3}{2} \cdot \frac{x^{-1}}{x^{-3}}=0$.
- B. $\lim _{x \rightarrow 0^{+}} \frac{\ln x}{\ln \sin x}=\lim _{x \rightarrow 0^{+}} \frac{x^{-1}}{\cot x}=\lim _{x \rightarrow 0^{+}} \frac{\tan x}{x}=1$.
- C. $\lim _{x \rightarrow 0}(\frac{1}{x}-\frac{1}{e^{x}-1})=\lim _{x \rightarrow 0} \frac{e^{x}-1-x}{x e^{x}-x}=\lim _{x \rightarrow 0} \frac{e^{x}-1}{e^{x}-1+x e^{x}}=\lim _{x \rightarrow 0} \frac{e^{x}}{2 e^{x}+x e^{x}}=\frac{1}{2}$.
- D. $\lim _{x \rightarrow 1} \frac{x^{3}-3 x+2}{x^{3}-x^{2}-x+1}=\lim _{x \rightarrow 1} \frac{3 x^{2}-3}{3 x^{2}-2 x-1}=\lim _{x \rightarrow 1} \frac{6 x}{6 x-2}=\lim _{x \rightarrow 1} \frac{6}{6}=1$.
题目解答
答案
**答案：A, B, C, D**
**解析：**
- **选项A：**
\[
\lim_{x \to 0^+} 3x^2 \ln x = \lim_{x \to 0^+} 3 \cdot \frac{\ln x}{x^{-2}} \stackrel{\text{洛必达}}{=} \lim_{x \to 0^+} -\frac{3x^3}{2} = 0
\]
正确。
- **选项B：**
\[
\lim_{x \to 0^+} \frac{\ln x}{\ln \sin x} \stackrel{\text{洛必达}}{=} \lim_{x \to 0^+} \frac{\sin x}{x \cos x} = 1 = \lim_{x \to 0^+} \frac{\tan x}{x}
\]
正确。
- **选项C：**
\[
\lim_{x \to 0} \left( \frac{1}{x} - \frac{1}{e^x - 1} \right) \stackrel{\text{通分}}{=} \lim_{x \to 0} \frac{e^x - 1 - x}{x(e^x - 1)} \stackrel{\text{洛必达两次}}{=} \frac{1}{2}
\]
正确。
- **选项D：**
\[
\lim_{x \to 1} \frac{x^3 - 3x + 2}{x^3 - x^2 - x + 1} \stackrel{\text{洛必达两次}}{=} \frac{3}{2} \quad \text{或} \quad \lim_{x \to 1} \frac{(x-1)^2(x+2)}{(x-1)^2(x+1)} = \frac{3}{2}
\]
正确。
**答案：A, B, C, D**