<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass{hfutexam}
\newcommand{\diff}{\,\mathrm{d}} % å®šä¹‰å¾®åˆ†ç¬¦å·

\begin{document}
\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´2æœˆ29æ—¥8:00-10:00}
\MingTiJiaoShi{é›†ä½“}
% \XiZhuRenQianMing[4em]{dengbing.png}

\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{\normalsize 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°.
% è‡ªåŠ¨æ&nbsp;¹æ®é€‰é¡¹é•¿åº¦è®¾ç½®è¡Œæ•°
\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 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‰(~~~~).
% æ‰‹åŠ¨è®¾ç½®ä¸ºæ¯è¡Œ1ä¸ª
\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}
\newpage
\item è®¾å‡½æ•° $f(x)$ 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•° $f(x)=\ln x+\dfrac2{x^2}, x\in(0,+\infty)$. 
\begin{enumerate}
\item[(1)] å‡½æ•° $f(x)$ çš„å¢žå‡åŒºé—´åŠæžå€¼;
\item[(2)] æ›²çº¿ $y=f(x)$ çš„å‡¹å‡¸åŒºé—´åŠæ‹ç‚¹.
\end{enumerate}
\end{enumerate}

\end{document}




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