单选题 （共 3 题 ），每题只有一个选项正确
(1) 设 $f(x)$ 满足 $\lim _{x \rightarrow 0} \frac{\sqrt{1+f(x) \sin 2 x}-1}{e^{x^2}-1}=1$, 则( )
$\text{A.}$ $f(0)=0$
$\text{B.}$ $\lim _{x \rightarrow 0} f(x)=0$
$\text{C.}$ $f^{\prime}(0)=1$
$\text{D.}$ $\lim _{x \rightarrow 0} f^{\prime}(x)=1$
设 $\alpha_1=\sqrt{1+\tan x}-\sqrt{1+\sin x}, \alpha_2=\int_0^{x^4} \frac{1}{\sqrt{1-t^2}} d t, \alpha_3=\int_0^x d u \int_0^{u^2} \arctan t d t$. 当 $x \rightarrow 0$ 时, 以上 3 个无穷小量按照从低阶到高阶的排序是（）
$\text{A.}$ $\alpha_1, \alpha_2, \alpha_3$.
$\text{B.}$ $\alpha_1, \alpha_3, \alpha_2$.
$\text{C.}$ $\alpha_2, \alpha_1, \alpha_3$.
$\text{D.}$ $\alpha_3, \alpha_1, \alpha_2$.
若 $\lim _{n \rightarrow \infty} \frac{n^a}{(n+1)^b-n^b}=2022$, 则
$\text{A.}$ $a=-\frac{2021}{2022}, b=\frac{1}{2022}$.
$\text{B.}$ $a=\frac{2021}{2022}, b=-\frac{1}{2022}$.
$\text{C.}$ $a=\frac{2021}{2022}, b=\frac{1}{2022}$.
$\text{D.}$ $a=-\frac{2021}{2022}, b=-\frac{1}{2022}$.