<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[simple]{hfutexam}
\usetikzlibrary{arrows.meta}
\RequirePackage{extarrows} % ç”¨äºŽç­‰å·ä¸Šé¢åŠ&nbsp;æ–‡å­—
\newcommand{\diff}{\,\mathrm{d}} % å®šä¹‰å¾®åˆ†ç¬¦å·

\begin{document}

% \tableofcontents
\BiaoTi{åˆè‚¥å·¥ä¸šå¤§å­¦æœŸä¸­è¯•å·}
\XueNian{2021}{2022}
\XueQi{äºŒ}
\KeChengDaiMa{034Y01}
\KeChengMingCheng{æ•°å­¦ï¼ˆä¸‹ï¼‰}
\XueFen{5}
\KeChengXingZhi{å¿…ä¿®}
\KaoShiXingShi{é—­å·}
\ZhuanYeBanJi{å°‘æ•°æ°‘æ—é¢„ç§‘ç­}
\KaoShiRiQi{2022å¹´5æœˆ13æ—¥8:00-10:00}
\MingTiJiaoShi{é›†ä½“}
\maketitle


\begin{enumerate}
\item \textbf{(10åˆ†)} æ±‚å‡½æ•° $\displaystyle f(x)=\ln\frac1{\sqrt{x^2-1}}+\arctan\frac1x$ çš„å®šä¹‰åŸŸ.
\item \textbf{(5åˆ†)} æ±‚å‡½æ•° $\displaystyle y=\begin{cases}
1/x,&amp;x&lt;0,\\1,&amp;x=0,\\1+e^{-x},&amp;x&gt;0\end{cases}$ çš„åå‡½æ•°.
\item \textbf{(10åˆ†)} æ±‚æžé™ $\displaystyle\lim_{x\to0^-}(1-x)^{1/x}$.
\item \textbf{(5åˆ†)} æ±‚æžé™ $\displaystyle\lim_{x\to-2}\frac{x^2-4}{x^3+8}$.
\item \textbf{(5åˆ†)} æ±‚æžé™ $\displaystyle\lim_{x\to0}\frac{\sin(e^{-x}-1)}{\arctan(1-\cos x)}$.
\item \textbf{(5åˆ†)} æ±‚æžé™ $\displaystyle\lim_{x\to0}\frac{\sqrt{1+2x-x^2}-\sqrt{1-2x+x^2}}x$. 
\item \textbf{(5åˆ†)} æ±‚æžé™ $\displaystyle\lim_{x\to\infty}\left(\cos\frac1x\right)^{\frac1{\ln(1+x^2)-2\ln x}}$.
\item \textbf{(5åˆ†)} æ±‚æžé™ $\displaystyle\lim_{x\to\infty}\left(\frac\pi{e^x-1}-\arctan\frac x2\right)$. 
\item \textbf{(5åˆ†)} æ±‚æžé™ $\displaystyle\lim_{n\to\infty}\left(\frac1{n^2+2}+\frac2{n^2+4}+\cdots+\frac n{n^2+2n}\right)$.
\item \textbf{(5åˆ†)} è®¾ $a_1=4,a_{n+1}=\sqrt{a_n+6}$, è¯æ˜Ž $\displaystyle\lim_{n\to\infty}a_n$ å­˜åœ¨å¹¶æ±‚ä¹‹.
\item \textbf{(10åˆ†)} è¯æ˜Ž $e^x+x=4$ åœ¨ $(0,+\infty)$ å†…æœ‰é›¶ç‚¹.
\item \textbf{(5åˆ†)} è®¾å‡½æ•° $f(x)$ åœ¨ $[-1,1]$ ä¸Šè¿žç»­, ä¸” $f(-1)\le1\le f(1)$. è¯æ˜Žå­˜åœ¨ $\xi\in[-1,1]$, ä½¿å¾— $f(\xi)=\xi^2$.
\item \textbf{(10åˆ†)} æ±‚ $y=e^{x+1}\sin x-e^2\sin1$ çš„å¯¼æ•°.
\item \textbf{(5åˆ†)} æ±‚ $y=\arctan e^x$ çš„å¯¼æ•°.
\item \textbf{(5åˆ†)} æ±‚æ›²çº¿ $y=\tan x$ åœ¨ç‚¹ $\left(-\dfrac\pi4,-1\right)$ å¤„çš„åˆ‡çº¿æ–¹ç¨‹å’Œæ³•çº¿æ–¹ç¨‹.
\item \textbf{(5åˆ†)} è®¾
$\displaystyle f(x)=\begin{cases}\dfrac{e^{3x}-1}{\arctan x},&amp;x&lt;0,\\2x+a,&amp;x\ge0\end{cases}$
åœ¨ $x=0$ å¤„è¿žç»­, æ±‚å¸¸æ•° $a$.
\end{enumerate}
\newpage

\BiaoTi{åˆè‚¥å·¥ä¸šå¤§å­¦è¯•å·ï¼ˆAï¼‰}
\XueNian{2021}{2022}
\XueQi{äºŒ}
\KeChengDaiMa{034Y01}
\KeChengMingCheng{æ•°å­¦ï¼ˆä¸‹ï¼‰}
\XueFen{5}
\KeChengXingZhi{å¿…ä¿®}
\KaoShiXingShi{é—­å·}
\ZhuanYeBanJi{å°‘æ•°æ°‘æ—é¢„ç§‘ç­}
\KaoShiRiQi{2022å¹´6æœˆ18æ—¥8:00-10:00}
\MingTiJiaoShi{é›†ä½“}
\maketitle

\tigan{ä¸€ã€å¡«ç©ºé¢˜ï¼ˆæ¯é¢˜3åˆ†ï¼Œå…±18åˆ†ï¼‰}
\begin{enumerate}
\item å¦‚æžœ $f(x)&gt;0$ ä¸” $\displaystyle\lim_{x\to\infty}f(x)=0$, åˆ™ $\displaystyle\lim_{x\to\infty}\bigl[1+f(x)\bigr]^{1/f(x)}=$\fillblank{}.
\item è®¾ $y=\sin(x^2+1)$, åˆ™ $\diff y=$\fillblank{}.
\item æžé™ $\displaystyle\lim_{n\to\infty}\left(\frac1{n^2-1}+\frac2{n^2-2}+\cdots+\frac n{n^2-n}\right)=$\fillblank{}.
\item æ›²çº¿ $y=2\ln(x+1)$ åœ¨ç‚¹ $(1,2\ln2)$ å¤„çš„åˆ‡çº¿æ–¹ç¨‹ä¸º\fillblank{}.
\item è‹¥ $e^{y-1}=1+xy$, åˆ™ $\dfrac{\diff y}{\diff x}\bigg|_{x=0}=$\fillblank{}.
\item å¦‚æžœå‡½æ•° $f(x)$ çš„å®šä¹‰åŸŸæ˜¯ $(0,+\infty)$, ä¸” $x=0$ æ˜¯æ›²çº¿ $y=f(x)$ çš„åž‚ç›´æ¸è¿‘çº¿, é‚£ä¹ˆ $\displaystyle\lim_{x\to0^+}\frac1{f(x)}=$\fillblank{}.
\end{enumerate}

\tigan{äºŒã€é€‰æ‹©é¢˜ï¼ˆæ¯é¢˜3åˆ†ï¼Œå…±18åˆ†ï¼‰}
\begin{enumerate}
\item å½“ $x\to+\infty$ æ—¶, $\dfrac1x$ å’Œ(~~~~)æ˜¯ç­‰ä»·æ—&nbsp;ç©·å°.
\xx{$\sin\dfrac1x$}{$\sin x$}{$e^{-x}$}{$e^{1/x}$}
\item è‹¥å½“ $x\to0$ æ—¶, $\arctan(e^x-1)\cdot(\cos x-1)$ å’Œ $x^n$ æ˜¯åŒé˜¶æ—&nbsp;ç©·å°, åˆ™ $n=$(~~~~).
\xx{$0$}{$1$}{$2$}{$3$}
\item è®¾ $f(x)=\arctan\dfrac1{x(x-1)^2}$, åˆ™ $x=0$ æ˜¯ $f(x)$ çš„(~~~~).
\xx{å¯åŽ»é—´æ–­ç‚¹}{è·³è·ƒé—´æ–­ç‚¹}{ç¬¬äºŒç±»é—´æ–­ç‚¹}{è¿žç»­ç‚¹}
\item 
\begin{tikzpicture}[overlay,xshift=12cm,yshift=-3cm]
\draw[-Stealth,thick](-3,0)--(3,0);
\draw[-Stealth,thick](0,-0.8)--(0,2.5);
\draw[very thick,smooth,domain=-55:55] plot ({\x/50-1.3}, {tan(\x)*tan(\x)});
\draw[very thick,smooth,domain=0.15:2] plot ({\x}, {-ln(\x)});
\draw
	(-0.3,-0.3) node {$O$}
	(2.8,-0.3) node {$x$}
	(-0.3,2.3) node {$y$};
\end{tikzpicture}
è®¾ $f(x)$ æ˜¯å®šä¹‰åœ¨ $(-\infty,+\infty)$ ä¸Šçš„è¿žç»­å‡½æ•°, ä¸” $f'(x)$ çš„å›¾åƒå¦‚ä¸‹å›¾æ‰€ç¤º, åˆ™ $f(x)$ æœ‰(~~~~).
\xx[1]{ä¸€ä¸ªæžå¤§å€¼ç‚¹ï¼Œæ²¡æœ‰æžå°å€¼ç‚¹}{æ²¡æœ‰æžå¤§å€¼ç‚¹ï¼Œä¸€ä¸ªæžå°å€¼ç‚¹}{ä¸€ä¸ªæžå¤§å€¼ç‚¹å’Œä¸€ä¸ªæžå°å€¼ç‚¹}{ä¸€ä¸ªæžå¤§å€¼ç‚¹å’Œä¸¤ä¸ªæžå°å€¼ç‚¹}
\item è®¾ $f(x)$ åœ¨ç‚¹ $x=0$ å¤„å¯å¯¼, ä¸” $f(0)=0$, åˆ™ $\displaystyle\lim_{x\to0}\frac{f(x^{2022})+x^{2021}f(x)}{x^{2022}}=$(~~~~).
\xx{$0$}{$f'(0)$}{$2f'(0)$}{$2022f'(0)$}
\item å¦‚æžœç‚¹ $(x_0,y_0)$ æ˜¯æ›²çº¿ $y=f(x)$ çš„æ‹ç‚¹, åˆ™ $f''(x_0)=$(~~~~).
\xx{$0$}{$\infty$}{ä¸å­˜åœ¨}{$0$ æˆ–ä¸å­˜åœ¨}
\end{enumerate}

\tigan{ä¸‰ã€è§£ç­”é¢˜ï¼ˆæ¯é¢˜8åˆ†ï¼Œå…±64åˆ†ï¼‰}
\begin{enumerate}
\item æ±‚æžé™ $\displaystyle\lim_{x\to-1}\frac{x^2-1}{x^2+3x+2}$.
\item æ±‚æžé™ $\displaystyle\lim_{x\to0}\frac{e^x-1-x}{\arcsin x^2}$.
\item è®¾ $\begin{cases}x=t^2+t&amp;\\y=t^3+t&amp;\end{cases}$, æ±‚ $\dfrac{\diff y}{\diff x}$ å’Œ $\dfrac{\diff^2 y}{\diff x^2}$.
\item è®¾ $f(x)=\begin{cases}x\arctan\dfrac1x,&amp;x&lt;0,\\x^2+ax+b,&amp;x\ge0.\end{cases}$
æ±‚å¸¸æ•° $a,b$ ä½¿å¾—å‡½æ•° $f(x)$ åœ¨ $(-\infty,+\infty)$ å†…å¯å¯¼, å¹¶æ±‚å‡ºæ­¤æ—¶æ›²çº¿ $y=f(x)$ çš„æ¸è¿‘çº¿.
\item	æ±‚å‡½æ•° $f(x)=x^3-x^2-x$ åœ¨åŒºé—´ $[-2,2]$ ä¸Šçš„æœ€å¤§å€¼å’Œæœ€å°å€¼.
\item è¯æ˜Ž: å½“ $-\dfrac\pi2&lt;x_1&lt;x_2&lt;\dfrac\pi2$ æ—¶, $\tan x_2-\tan x_1\ge x_2-x_1$.
\item è®¾å‡½æ•° $f(x)$ åœ¨ $(-\infty,+\infty)$ å†…å¯å¯¼, ä¸” $f(1)=0$.
è¯æ˜Ž: å­˜åœ¨ $\xi\in(0,1)$ ä½¿å¾— $\xi f'(\xi)+2022f(\xi)=0$.
\item è®¾å‡½æ•° $f(x)=\ln x+\dfrac2{x^2}, x\in(0,+\infty)$. æ±‚
\begin{enumerate}
\item[(1)] å‡½æ•° $f(x)$ çš„å¢žå‡åŒºé—´åŠæžå€¼ï¼›
\item[(2)] æ›²çº¿ $y=f(x)$ çš„å‡¹å‡¸åŒºé—´åŠæ‹ç‚¹.
\end{enumerate}
\end{enumerate}

\newpage
\BiaoTi{åˆè‚¥å·¥ä¸šå¤§å­¦è¯•å·å‚è€ƒç­”æ¡ˆï¼ˆAï¼‰}
\XueNian{2021}{2022}
\XueQi{äºŒ}
\KeChengDaiMa{034Y01}
\KeChengMingCheng{æ•°å­¦ï¼ˆä¸‹ï¼‰}
\XueFen{5}
\KeChengXingZhi{å¿…ä¿®}
\KaoShiXingShi{é—­å·}
\ZhuanYeBanJi{å°‘æ•°æ°‘æ—é¢„ç§‘ç­}
\KaoShiRiQi{2022å¹´6æœˆ18æ—¥8:00-10:00}
\MingTiJiaoShi{é›†ä½“}
\maketitle

\tigan{ä¸€ã€å¡«ç©ºé¢˜ï¼ˆæ¯å°é¢˜3åˆ†ï¼Œå…±18åˆ†ï¼‰}

\textbf{è¯·å°†ä½&nbsp;çš„ç­”æ¡ˆå¯¹åº”å¡«åœ¨æ¨ªçº¿ä¸Šï¼š}

\textbf{1.} \fillblank{$e$}, 
\textbf{2.} \fillblank{$2x\cos(x^2+1)\diff x$}, 
\textbf{3.} \fillblank{$\dfrac12$}, \\
\textbf{4.} \fillblank{$y=x-1+2\ln 2$},
\textbf{5.} \fillblank{$1$},
\textbf{6.} \fillblank{$0$}.

\tigan{äºŒã€é€‰æ‹©é¢˜ï¼ˆæ¯å°é¢˜3åˆ†ï¼Œå…±18åˆ†ï¼‰}

\textbf{è¯·å°†ä½&nbsp;æ‰€é€‰æ‹©çš„å­—æ¯ A, B, C, D ä¹‹ä¸€å¯¹åº”å¡«åœ¨ä¸‹åˆ—è¡¨æ&nbsp;¼é‡Œï¼š}

\xuanzeti{\textbf{é¢˜å·}}{\textbf{ç­”æ¡ˆ}}%
\xuanzeti{1}{A}%
\xuanzeti{2}{D}%
\xuanzeti{3}{B}%
\xuanzeti{4}{A}%
\xuanzeti{5}{C}%
\xuanzeti{6}{D}

\tigan{ä¸‰ã€è§£ç­”é¢˜ï¼ˆæ¯å°é¢˜8åˆ†ï¼Œå…±64åˆ†ï¼‰}

\textbf{1. ï¼ˆ8åˆ†ï¼‰ã€è§£ã€‘}
\vspace{-\baselineskip}

\begin{align*}
\lim_{x\to-1}\frac{x^2-1}{x^2+3x+2}&amp;=\lim_{x\to-1}\frac{(x-1)(x+1)}{(x+2)(x+1)} \score3\\
&amp;=\lim_{x\to-1}\frac{x-1}{x+2} \score3\\
&amp;=\frac{-2}1=-2. \score2
\end{align*}

\textbf{2. ï¼ˆ8åˆ†ï¼‰ã€è§£ã€‘}
\vspace{-\baselineskip}

\begin{align*}
\lim_{x\to0}\frac{e^x-1-x}{\arcsin x^2}&amp;=\lim_{x\to0}\frac{e^x-1-x}{x^2} \score3\\
&amp;\xlongequal[]{\text{æ´›å¿…è¾¾}}\lim_{x\to0}\frac{e^x-1}{2x} \score3\\
&amp;=\lim_{x\to0}\frac{x}{2x}=\frac12. \score2
\end{align*}

\textbf{3. ï¼ˆ8åˆ†ï¼‰ã€è§£ã€‘}
\vspace{-\baselineskip}

\begin{align*}
\frac{\diff y}{\diff x}&amp;=\frac{\diff y/\diff t}{\diff x/\diff t} \score2\\
&amp;=\frac{3t^2+1}{2t+1}, \score2\\
\frac{\diff^2 y}{\diff x^2}&amp;=\frac{\diff y'/\diff t}{\diff x/\diff t} \score2\\
&amp;=\frac{6t(2t+1)-(3t^2+1)2}{(2t+1)^3}=\frac{6t^2+6t-2}{(2t+1)^3}. \score2
\end{align*}

\newpage
\textbf{4. ï¼ˆ8åˆ†ï¼‰ã€è§£ã€‘}

\indent ç”±äºŽ $f(x)$ åœ¨ $x=0$ å¤„è¿žç»­, å›&nbsp;æ­¤
\begin{align*}
f(0)&amp;=f(0^+) \score1\\
&amp;=b=\lim_{x\to0^-}x\arctan\frac1x=0\times\left(-\frac\pi2\right)=0. \score1
\end{align*}

\indent ç”±äºŽ $f(x)$ åœ¨ $x=0$ å¤„å¯å¯¼, å›&nbsp;æ­¤
\begin{align*}
f'_-(0)&amp;=f'_+(0), \score1\\
f'_-(0)&amp;=\lim_{x\to0^-}\frac{x\arctan\frac1x}x=\lim_{x\to0^-}\arctan\frac1x=-\frac\pi2 \score1\\
f'_+(0)&amp;=(2x+a)|_{x=0}=a, \score1
\end{align*}
å›&nbsp;æ­¤ $a=-\dfrac\pi2$. \score1

\indent ç”±äºŽ
\begin{align*}
\lim_{x\to+\infty}\frac yx&amp;=\lim_{x\to+\infty}\left(x-\frac\pi2\right)=+\infty, \score1\\
\lim_{x\to-\infty}\frac yx&amp;=\lim_{x\to-\infty}\arctan\frac1x=0,\\
\lim_{x\to-\infty}y&amp;=\lim_{x\to-\infty}x\arctan\frac1x=\lim_{t\to0^-}\frac{\arctan t}t=1,
\end{align*}
å›&nbsp;æ­¤æ›²çº¿ $y=f(x)$ çš„æ¸è¿‘çº¿åªæœ‰ $y=1$. \score1

\textbf{5. ï¼ˆ8åˆ†ï¼‰ã€è§£ã€‘}

\indent ç”±
\[f'(x)=3x^2-2x-1=(3x+1)(x-1)=0 \score2\]
å¯å¾—é©»ç‚¹ $x=-\dfrac13,1$. \score2

\indent ç”±äºŽ
\[f(-2)=-10,\quad f(2)=2,\quad f\left(-\frac13\right)=\frac5{27},\quad f(1)=-1, \score2\]
å›&nbsp;æ­¤æœ€å¤§å€¼ä¸º $2$, æœ€å°å€¼ä¸º $-10$. \score2

\textbf{6. ï¼ˆ8åˆ†ï¼‰ã€è¯æ˜Žã€‘}

\textbf{è¯æ³•ä¸€}: è®¾ $f(x)=\tan x-x$, åˆ™ \score2
\[f'(x)=\frac1{\cos^2x}-1=\tan^2x\ge0. \score2\]
å›&nbsp;æ­¤ $f(x)$ åœ¨ $\left(-\dfrac\pi2,\dfrac\pi2\right)$ ä¸Šå•è°ƒé€’å¢ž, ä»Žè€Œ \score2
\[f(x_2)\ge f(x_1),\quad\tan x_2-\tan x_1\ge x_2-x_1. \score2\]

\newpage
\textbf{è¯æ³•äºŒ}: è®¾ $f(x)=\tan x$, åˆ™ $f(x)$ åœ¨ $[x_1,x_2]$ ä¸Šè¿žç»­, $(x_1,x_2)$ å†…å¯å¯¼. \score2

\indent ç”±æ‹‰æ&nbsp;¼æœ—æ—¥ä¸­å€¼å®šç†, å­˜åœ¨ $\xi\in(x_1,x_2)$ ä½¿å¾—
\[\frac{f(x_2)-f(x_1)}{x_2-x_1}=f'(\xi), \score2\]
å³
\[\frac{\tan x_2-\tan x_1}{x_2-x_1}=\frac1{\cos^2\xi}\ge1. \score2\]
æ‰€ä»¥ $\tan x_2-\tan x_1\ge x_2-x_1$. \score2

\textbf{7. ï¼ˆ8åˆ†ï¼‰ã€è¯æ˜Žã€‘}

\indent è®¾ $F(x)=x^{2022}f(x)$, \score2\\
åˆ™ $F(x)$ åœ¨ $[0,1]$ ä¸Šè¿žç»­, $(0,1)$ å†…å¯å¯¼, \score1\\
ä¸” $F(0)=0,F(1)=f(1)=0$. \score1

\indent ç”±ç½—å°”ä¸­å€¼å®šç†, å­˜åœ¨ $\xi\in(0,1)$ ä½¿å¾— $F'(\xi)=0$. \score2\\
ç”±äºŽ $F'(x)=x^{2022}f'(x)+2022x^{2021}f(x)$ ä¸” $\xi\neq0$, \score1\\
æ‰€ä»¥ $\xi f'(\xi)+2022f(\xi)=1$. \score1


\textbf{8. ï¼ˆ8åˆ†ï¼‰ã€è§£ã€‘}

(1) 
\[f'(x)=\frac1x-\frac4{x^3}=\frac{x^2-4}{x^3}=\frac{(x+2)(x-2)}{x^3}. \score1\]
å½“ $0&lt;x&lt;2$ æ—¶, $f'(x)&lt;0$. å½“ $x&gt;2$ æ—¶, $f'(x)&gt;0$. \score1\\
å›&nbsp;æ­¤ $(0,2]$ æ˜¯ $f(x)$ çš„å•å‡åŒºé—´,\\
$[2,+\infty)$ æ˜¯ $f(x)$ çš„å•å¢žåŒºé—´. \Score{(1åˆ†, å†™æˆå¼€åŒºé—´ä¸æ‰£åˆ†)}\\
æ‰€ä»¥ $f(x)$ åªæœ‰å”¯ä¸€çš„æžå°å€¼ $f(2)=\ln2+\dfrac12$. \score1

(2) 
\[f''(x)=-\frac1{x^2}+\frac{12}{x^4}=-\frac{x^2-12}{x^4}=-\frac{(x-2\sqrt3)(x+2\sqrt3)}{x^4}. \score1\]
å½“ $0&lt;x&lt;2\sqrt3$ æ—¶, $f''(x)&gt;0$. å½“ $x&gt;2\sqrt3$ æ—¶, $f''(x)&lt;0$. \score1\\
å›&nbsp;æ­¤ $(0,2\sqrt3]$ æ˜¯æ›²çº¿ $y=f(x)$ çš„å‡¹åŒºé—´,\\
$[2\sqrt3,+\infty)$ æ˜¯æ›²çº¿ $y=f(x)$ çš„å‡¸åŒºé—´,\Score{(1åˆ†, å†™æˆå¼€åŒºé—´ä¸æ‰£åˆ†)}\\
æ‹ç‚¹ä¸º $\left(2\sqrt3,\ln(2\sqrt3)+\dfrac16\right)$. \score1

\end{document}




</pre></body></html>