考研数学训练题——高等数学1极限与连续

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考研数学复习教程答案详解高数部分

第一篇高等数学第一章函数、极限与连续强化训练（一）一、选择题1.2.提示：参照“例1.1.5”求解。

3.4.解因选项(D)中的不能保证任意小，故选(D)5.6.7.8.9.10.二、填空题11.提示：由2cos 12sin 2xx =-可得。

12.13.提示：由1 未定式结果可得。

14.提示：分子有理化，再同除以n即可。

15.提示：分子、分母利用等价无穷小代换处理即可。

16.17.提示：先指数对数化，再利用洛必达法则。

18.19.解因()2000122(1cos )22cos 2lim lim lim lim lim 1x x x x x x x xx f x x xxx -----→→→→→⋅---=====- ()0lim lim xx x f x ae a --→→==，而()0f a =，故由()f x 在 0x =处连续可知，1a =-。

20.提示：先求极限（1∞型）得到()f x 的表达式，再求函数的连续区间。

三、解答题 21.(1)(2)提示：利用皮亚诺型余项泰勒公式处理12sin ,sin x x。

(3)(4)(5)提示：先指数对数化，再用洛必达法则。

(6)提示：请参照“例1.2.14（3）”求解。

22.23.解由题设极限等式条件得21()ln(cos )201()lim ,limln(cos )1f x x xxx x f x e e x x x+→→=+=，即 2201()1()limln(cos )lim ln(1cos 1)1x x f x f x x x x x x x→→+=+-+=，利用等价无穷小代换，得201()lim(cos 1)1x f x x x x →-+=，即230cos 1()lim()1x x f x x x→-+=，故 30()3lim 2x f x x →=。

24.提示：先指数对数化，再由导数定义可得。

25.26.28.提示：利用皮亚诺型余项泰勒公式求解。

考研高数极限试题及答案

考研高数极限试题及答案模拟试题：一、选择题（每题3分，共15分）1. 极限 $\lim_{x \to 0} \frac{\sin x}{x}$ 的值是多少？A. 0B. 1C. -1D. $\frac{1}{2}$2. 函数 $f(x) = \frac{x^2 - 1}{x - 1}$ 在 $x = 1$ 处的极限是多少？A. 2B. 1C. 0D. 不存在3. 极限 $\lim_{x \to +\infty} \frac{x^2}{e^x}$ 存在吗？A. 是B. 否4. 函数 $g(x) = \begin{cases}x^2 & \text{if } x \neq 0 \\0 & \text{if } x = 0\end{cases}$ 在 $x = 0$ 处的右极限是多少？A. 0B. 1C. $\frac{1}{2}$D. 不存在5. 极限 $\lim_{x \to 1} (x^2 - 1)$ 等于多少？A. 0B. 1C. 2D. 3二、计算题（每题10分，共40分）6. 计算极限 $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$。

7. 计算极限 $\lim_{x \to 0} \frac{\cos x - 1}{x}$。

8. 计算极限 $\lim_{x \to +\infty} \frac{\sin x}{x}$。

9. 计算极限 $\lim_{n \to \infty} \frac{1}{n^2} +\frac{1}{n^3}$。

三、解答题（每题20分，共40分）10. 证明 $\lim_{x \to 0} x \sin \frac{1}{x} = 0$。

11. 已知 $\lim_{x \to 2} f(x) = 3$，证明 $\lim_{x \to 2} [f(x)]^2 = 9$。

极限与连续练习题及解析

极限与连续练习题及解析在数学课上，我们经常会遇到一些有关于极限与连续的练习题。

这些题目不仅能够帮助我们巩固对极限与连续的理解，还能提高我们解决问题的能力。

在本文中，我将为大家分享一些关于极限与连续的练习题及解析。

题目一：计算极限求解以下极限：1. $$\lim_{x\to 2}\frac{x^2-4}{x-2}$$解析：将被除数进行因式分解得：$$\lim_{x\to 2}\frac{(x+2) \cdot (x-2)}{x-2}$$最后得到：$$\lim_{x\to 2}(x+2)$$代入极限的定义，得到结果为：$$4$$题目二：证明函数连续证明下列函数在指定区间上连续：1. 函数$f(x)=\sqrt{x}$在区间$[0, +\infty)$上连续。

首先，我们需要证明$f(x)=\sqrt{x}$在$[0, +\infty)$上存在。

由于$x \geq 0$，所以$\sqrt{x}$是有定义的。

接下来，我们需要证明对于任意给定的$\varepsilon > 0$，存在一个$\delta > 0$，使得当$0 < |x-a| <\delta$时，$|\sqrt{x}-\sqrt{a}|<\varepsilon$。

根据不等式$|\sqrt{x}-\sqrt{a}|<|\sqrt{x}+\sqrt{a}|$，可以得到$$|\sqrt{x}-\sqrt{a}|<|\sqrt{x}-\sqrt{a}|\cdot\frac{|\sqrt{x}+\sqrt{a}|}{|\sqrt{x}-\sqrt{a}|}$$进一步化简得：$$|\sqrt{x}-\sqrt{a}|<\frac{|\sqrt{x}^2-\sqrt{a}^2|}{|\sqrt{x}-\sqrt{a}|}$$继续化简得：$$|\sqrt{x}-\sqrt{a}|<\frac{|x-a|}{|\sqrt{x}+\sqrt{a}|}$$由于$\sqrt{x}+\sqrt{a}$在$x$趋于$a$时不等于0，所以存在一个正数$M$，使得$|\sqrt{x}-\sqrt{a}|<M|x-a|$。

第二章-极限与连续--基础练习题(含解答)

第二章极限与连续基础练习题（作业）§2.1 数列的极限一、观察并写出下列数列的极限：1．4682,,,357极限为1 2．11111,,,,,2345--极限为03．212212⎧-⎪⎪=⎨+⎪⎪⎩n nn nnn a n 为奇数为偶数极限为1§2.2 函数的极限一、画出函数图形，并根据函数图形写出下列函数极限： 1．lim →-∞xx e极限为零 2．2lim tan x x π→无极限3．lim arctan →-∞x x极限为2π-4．0lim ln x x +→ 无极限，趋于-∞二、设2221,1()3,121,2x x f x x x x x x +⎧⎪=-+<⎨⎪->⎩，问当1x →，2x →时，()f x 的极限是否存在？211lim ()lim(3)3x x f x x x ++→→=-+=；11lim ()lim(21)3x x f x x --→→=+= 1lim () 3.x f x →∴=222lim ()lim(1)3x x f x x ++→→=-=；222lim ()lim(3)53x x f x x x --→→=-+=≠ 2lim ()x f x →∴不存在。

三、设()111xf x e=+，求 0x →时的左、右极限，并说明0x →时极限是否存在．()101lim lim 01x x xf x e ++→→==+()11lim lim 11x x x f x e--→→==+lim ()x f x →∴不存在。

四、试讨论下列函数在0x →时极限是否存在． 1．绝对值函数()||=f x x ，存在极限为零 2．取整函数()[]=f x x 不存在 3．符号函数()sgn =f x x 不存在§2.3 无穷小量与无穷大量一、判断对错并说明理由： 1．1sinx x是无穷小量．错，无穷小量需相对极限过程而言，在某个极限过程中的无穷小量在其它极限过程中可能不再是无穷小量。

考研数学一-高等数学函数极限连续(三)_真题-无答案

考研数学一-高等数学函数、极限、连续(三)(总分100,考试时间90分钟)Section Ⅰ Use of EnglishDirections:Read the following text. Choose the best word(s) for each numbered blank and mark A, B, C or D on ANSWER SHEET 1. (10 points)Attempts to understand the relationship between social behavior and health have their origin in history. Dubos (1969) suggested that primitive humans were closer to the animals (1) they, too, relied'upon their instincts to stay healthy. Yet some primitive humans (2) a cause and effect relationship between doing certain things and alleviating (3) of a disease or (4) the condition of a wound. (5) there was so much that primitive humans did not (6) the functioning of the body, magic became an **ponent ofthe beliefs about the causes and cures of heath (7) Therefore it is not (8) that early humans thought that illness was caused (9) evil spirit. Primitive medicines made from vegetables or animals were invariably used in combination with some form of ritual to (10) harmful spirit from a diseased body.One of the. earliest (11) in the Western world to formulate principles of health care based upon rational thought and (12) of supernatural phenomena is found in the work of the Greek physician Hippocrates. The writing (13) to him has provided a number of principles underiying modern medical practice. One of his most famous (14) , the Hippocratic Oath, is the foundation of contemporary medical ethics.Hippocrates also argued that medical knowledge should be derived from a (15) of the natural science and the logic of cause and effect relationships. In this (16) thesis, On Air, Water, and Places, Hippocrates pointed out that human well-being is (17) by the totality of environmental (18) : living habits or lifestyle, climate, geography of the land, and the quality of air, and food. (19) enough, concerns about our health and the quality of air, water, and places are (20) very much written in twentieth century.1. [A] in that [B] now that [C] because [D] so that2. [A] identified [B] recognized [C] admitted [D] esteemed3. [A] symbols [B] signs [C] symptoms [D] syndromes4. [A] arranging [B] adjusting [C] developing [D] improving5. [A] While [B] Although [C] Provided [D] Since6. [A] confirm [B] appreciate [C] tolerate [D] discover7. [A] damages [B] agitation [C] disorders [D] collapse8. [A] surprising [B] obvious [C] promising [D] unwise9. [A] in [B] by [C] off [D] up10. [A] discharge [B] exclude [C] dismiss [D] expel11. [A] expedition [B] incentives [C] stimuli [D] endeavors12. [A] foundation [B] rejection [C] integration [D] acceptance13. [A] attributed [B] committed [C] acknowledged [D] confined14. [A] attractions [B] subscriptions [C] contributions [D] functions15. [A] creation [B] expectation [C] perception [D] preference16. [A] durable [B] classic [C] thorough [D] immediate17. [A] impressed [B] surpassed [C] influenced [D] regarded18. [A] elements [B] factors [C] components [D] deposits19. [A] Interesting [B] Appealing [C] Demanding [D] Exaggerating20. [A] even [B] indeed [C] still [D] moreoverSection Ⅱ Reading ComprehensionPart ADirections:Read the following four texts. Answer the questions below each text by choosing A, B, C or D. Mark your answers on ANSWER SHEET 1. (40 points)Text 1The increase in global trade means that **panies cannot afford to make costly advertising mistakes if they want to be competitive.Understanding the language and culture of target markets in foreign countries is one of the keys to successful international marketing. Too **panies, however, have jumped into foreign markets with embarrassing results.Translation mistakes are at the heart of many blunders in international advertising.General Motors, the US auto manufacturer, got a costly lesson when it introduced its Chevrolet Nova to the Puerto Rican market. "Nova" is Latin for "new (star)" and means "star" in many languages, but in spoken Spanish it can sound like "nova", meaning "it doesn't go". Few people wanted to buy a car with that cursed meaning. When GM changed the name to Caribe, sales "picked up" dramatically.Marketing blunders have also been made by food and **panies. One American **pany's friendly "Jolly Green Giant" (for advertising vegetables) became something quite different when it was translated into Arabic as "Intimidating Green Ogre".When translated into German, Pepsi's popular slogan, "Come Alive with Pepsi" came out implying "Come Alive from the Grave". No wonder customers in Germany didn't rush out to buy Pepsi.Successful international marketing doesn't stop with good translations—other aspects of culture must be researched and understood if marketers are to avoid blunders.When marketers do not understand and appreciate the values, tastes, geography, climate, superstitions, religion, or economy of a culture, they fail to capture their target market.For example, an American designer tried to introduce a new perfume into the Latin American market but the product aroused little interest. The main reason was that the camellia used in it was traditionally used for funerals in many South American countries.Having awakened to the special nature of foreign advertising, companies are becoming much more conscientious in their translations and more sensitive to cultural distinctions.The best way to prevent errors is to hire professional translators who understand the target language and its idiomatic usage, or to use a technique called "back translation" to reduce the possibility of blunders.The process used one person to translate a message into the target language and another to translate it back. Effective translators aim to capture the overall message of an advertisement because a word-for-word duplication of the original rarely conveys the intended meaning and often causes misunderstandings.In designing advertisements for other countries, messages need to be short and simple.They should also avoid jokes, since what is considered funny in one part of the world may not be so humorous in another.21. The best title of this passage might be______.A. Culture Is Very Important in AdvertisingB. Avoid Cultural Misunderstanding between NationsC. **e Cultural Sock in Different CountriesD. Advertisements Reflect Various Life Styles22. What does the word "blunder" mean in this passage?A. hesitationB. mistakeC. stutterD. default23. Which of the following statements can be used to summarize the gist from Paragraph 3 to Paragraph 6?A. Cultural shocksB. Faulty translationsC. Avoid cultural oversightsD. Prevent blunders24. We can learn from the context in Paragraph 9 that the word "camellia" most probably mean______.A. an animal used in perfume for its smellB. a piece of fabric used both in perfume and at funeralsC. a flower used in perfume for its fragrance and used for funeralsD. an ornament used in perfume and at funerals25. One way to prevent errors in advertising in different countries is to______.A. fire the translators who don't know the target languageB. use the technique called "literal translation" to reduce the possibility of blundersC. avoid cultural oversights and avoid certain jokesD. explain in details when designing advertisement for other countriesText 2My dictionary defines subtle as "not immediately obvious; characterized by skill or ingenuity; clever; elusive; [even] insidious." Let us look at some concrete instances of this. (The very word itself is an example, as the letter" b" is silent in pronunciation.)"Language can be straight-forward and directly to the point, but sometimes that takes the fun out of it. On occasion, at least, one appreciates subtleties—often as the mark of a quick wit. This particularly is true of jokes that generally have a double meaning. For example: Awaitress received only three pennies for a tip. Nonplussed, she told the customer that those three pennies told a lot about him. He took the bait and asked what they revealed. "The first penny," she said, "tells me you are thrifty." The patron agreed. "what does the second penny say?" asked the customer. "It tells me you're a bachelor." "Right again," he replied, "And what does the third penny tell you?" "The third penny," responded the waitress, "tells me your daddy was a bachelor, too." How's that for a subtle punishment?Subtleties also can be used on occasion for a good putdown. For instance, one can say a certain man was a big gun of industry. "Yes," is the counter, "he was fired several times." In this category was Mark Twain' s caustic time bomb: "He was a good man—in the worst sense of the term."In our day of political correctness (sometimes called the tyranny of the minority), police seldom talk about suspects, but only about" persons of interest." I guess law enforcement does not want another lawsuit on its hands. Then, too, with the campaign against fat and fried foods, Kentucky Fried Chicken calls itself KFC, figuring that few will think of "fried" that way.The meaning of a word or phrase seems to change more rapidly today and unless one is "with it", a faux pas (失礼) can be committed. Such is the case with the term, "an exceptional child." Way back when, one would think that referred to an especially bright youngster, whereas today it indicates a handicapped youth. So, too, the word "primitive" virtually has been erased from our language and replaced with "earlier culture" and Indians are known as Native Americans.The world of advertising is a master at subtleties with which it hopes to bamboozle (欺骗) the customer. For instance, when asked how much a gallon of gasoline costs, the reply might be something like $2.25. Yet, one must add a penny to that as a 9/10 follows the price, making it, in effect, $2.26. Cereal prices have skyrocketed over the years, but **panies claim to have held the line by keeping the price the same. What many do, however, is reduce the number of ounces in the package.I leave the reader with the truism that subtlety, not brevity, is the soul of wit. Use it to win friends and influence people.26. By saying "the letter 'b' is silent in pronunciation", the author wants to indicate that ______[A] One should not omit the letter when spelling the word.[B] This word is different from other words in pronunciation.[C] The meaning of the word can be illustrated by the word itself.[D] The meaning of the word is implied in the word itself.27. According to Paragraph 2, the waitress plays tricks on the patron by ______[A] telling him the particular meanings of the three pennies.[B] informing him a pun which is intended to insult him indirectly.[C] using a double-meaning joke as a punishment for that few tips.[D] explaining to him the implied meaning of the three-pennies.28. The underlined word "putdown" (Line 1, Paragraph 3) most probably means ______[A] a pun which can be interpreted in both positive and negative ways.[B] a tone-twisting joke which intends to make the hearer puzzled.[C] a response by understanding something deliberately in another way.[D] an indirect way of saying something to make someone humiliated.29. Which of the following has nothing to do with political correctness?[A] Police now seldom talk about "suspects" but only about "persons of interest".[B] Burger King rebrands "Whaler" sandwich as "Big Fish" with the campaign "Save the Whales".[C] Kentucky Fried Chicken calls itself KFC with the campaign against fat and fried foods[D] "An exceptional child" indicates a handicapped youth and Indians are known as Native Americans.30. We can infer from the text that the author describe the word "subtlety" with ______[A] ambiguity.[B] apprehension.[C] antipathy.[D] approval.Text 3In the early 1960s Wilt Chamberlain was one of the only three players in the National Basketball Association (NBA) listed at over seven feet. If he had played last season, however, he would have been one of 42. The bodies playing major professional sports have changed dramatically over the years, and managers have been more than willing to adjust team uniforms to fit the growing numbers of bigger, longer frames.The trend in sports, though, may be obscuring an unrecognized reality: Americans have generally stopped growing. Though typically about two inches taller now than 140 years ago, today' s people—especially those born to families who have lived in the U. S. for many generations--apparently reached their limit in the early 1960s. And they aren't likely to get any taller. "In the general population to- day, at this genetic, environmental level, we've pretty much gone as far as we can go," says anthropologist William Cameron Chumlea of Wright State University. In the case of NBA players, their in- crease in height appears to result from the **mon practice of recruiting players from all over the world.Growth, which rarely continues beyond the age of 20, demands calories andnutrients—notably, protein—to feed expanding tissues. At the start of the 20th century, under-nutrition and childhood infections got in the way. But as diet and health improved, children and adolescents have, on average, increased in height by about an inch and a half every 20 years, a pattern known as the secular trend in height. Yet according to the Centers for Disease Control and Prevention, average height—5'9" for men, 5'4" for women—hasn't really changed since 1960.Genetically speaking, there are advantages to avoiding substantial height. During childbirth, larger babies have more difficulty passing through the birth canal. Moreover, even though humans have been upright for millions of years, our feet and back continue to struggle with bipedal posture and can- not easily withstand repeated strain imposed by oversize limbs. "There are some real constraints that are set by the genetic architecture of the individual organism," says anthropologist William Leonard of Northwestern University.Genetic maximums can change, but don't expect this to happen soon. Claire C. Gordon, senior anthropologist at the Army Research Center in Natick, Mass. , ensures that 90 percent of the uniforms and workstations fit recruits without alteration. She says that, unlike those for basketball, the length of military uniforms has not changed for some time. And if you need to predict human height in the near future to design a piece of equipment, Gordon says that by and large, "you could use today's data and feel fairly confident.\31. Wilt Chamberlain is cited as an example to[A] illustrate the change of height of NBA players.[B] show the popularity of NBA players in the U.S.[C] compare different generations of NBA players.[D] assess the achievements of famous NBA players.32. Which of the following plays a key role in body growth according to the text?[A] Genetic modification.[B] Natural environment.[C] Living standards.[D] Daily exercise.33. On which of the following statements would the author most probably agree?[A] Non-Americans add to the average height of the nation.[B] Human height is conditioned by the upright posture.[C] Americans are the tallest on average in the world.[D] Larger babies tend to become taller in adulthood.34. We learn from the last paragraph that in the near future[A] the garment industry will reconsider the uniform size.[B] the design of military uniforms will remain unchanged.[C] genetic testing will be employed in selecting sportsmen.[D] the existing data of human height will still be applicable.35. The text intends to tell us that[A] the change of human height follows a cyclic pattern.[B] human height is becoming even more predictable.[C] Americans have reached their genetic growth limit.[D] the genetic pattern of Americans has altered.Text 4The Republican Party has lost its mind. To win elections, a party needs votes, obviously, and constituencies. First, however, it needs ideas. In 1994--95, the Republican Party had after long struggle advanced a coherent, compelling set of political ideas expressed in a specific legislative agenda. The political story of 1996 is that this same party, within the space of six weeks, then became totally, shockingly intellectually deranged.Then, astonishingly, on the very moment of their philosophical victory, just as the Republicans prepared to carry these ideas into battle in November, came cannon fire from the rear. Pat Buchanan first came out to declare a general insurrection. The enemy, according to Buchanan, is not the welfare state. It is that conservative icon, capitalism, with its ruthless captains of industry, greedy financiers and political elites (Republicans included, of course). All three groups collaborate to let foreigners--immigrants, traders, parasitic foreign-aid loafers--destroy the good life of the ordinary American worker.Buchananism would support and wield a big and mighty government apparatus to protect the little guy from buffeting, a government that builds trade walls and immigrant-repelling fences, that imposes punitive taxes on imports, and that polices the hiring and firing practices of business with the arrogance of the most zealous forcer.Republicans have focused too much on the mere tactical dangers posed by this assault. Yes, it gives ammunition to the Democrats. Yes, it puts the eventual nominee through a bruising campaign and delivers him tarnished and drained into the ring against Bill Clinton.But the real danger is philosophical, not tactical. It is axioms, not just policies, that are under fire. The Republican idea of smaller government is being ground to dust--by Republicans. In the middle of an election year, when they should be honing their themes against Democratic liberalism, Buchanan's rise is forcing a pointless rearguard battle against a philosophical corpse, the obsolete paleoconservatism--a mix of nativism, protectionism and isolationism--of the 1930s.As the candidates' debate in Arizona last week showed, the entire primary campaign will be fought on Buchanan's grounds, fending off his Smoot-Hawley-Franco populism. And then what? After the convention, what does the nominee do? Try to resurrect the anti-welfare state themes of the historically successful '94 congressional campaign?Political parties can survive bruising primary battles. They cannot survive ideological meltdown. Dole and Buchanan say they are fighting for the heart and soul of the Republican Party. Heart and soul, however, will get you nowhere when you've lost your way--and your mind.36. The writer believes that the Republic Party, in face of the challenge from its opponents, should[A] maintain its political ideas wisely.[B] protect a powerful government.[C] focus narrowly on strategies.[D] bring down rebellions at first.37. The word "deranged" (Para. 1) denotes[A] "zeal."[B] "hypocrisy."[C] "craze."[D] "bewildering.\38. Buchanan argues that the government has to hold in high regard[A] the survival of successive battles.[B] the uniformity of its political ideas.[C] the benefits of foreigners at large.[D] the interests of average Americans.39. Which of the following can best describe the eventual nominee of the Republic Party of the election?[A] "from a scratch."[B] "on the ropes."[C] "out of patience."[D] "under attack.\40. The passage shows that the writer is ______ the present situation.[A] surprised at[B] annoyed at[C] puzzled by[D] confused byPart BDirections:You are going to read a text about how to keep your job, followed by a list of important examples. Choose the best examples from the list A - F for each numbered subheading (41-45). There is one extra examples which you do not need to use. Mark your answers on ANSWER SHEET 1. (10 points)(41)______Through laziness and cowardice a large part of mankind, even after nature has freed them from alien guidance, gladly remain immature. It is because of laziness and cowardice that it is so easy for others to usurp the role of guardians, It is so comfortable to be a minor! If I have a book which provides meaning for me, a doctor who will judge my diet for me and so on, then I do not need to exert myself. I do not have any need to think; if I can pay, others will take over the tedious job for me. The guardians who have kindly undertaken the supervision will see to it that by far the largest part of mankind, including the entire "beautiful sex," should consider the step into maturity, not only as difficult but as very dangerous.(42)______It is difficult for the isolated individual to work himself out of the immaturity which has become almost natural for him. He has even become fond of it and for the time being is incapable of employing his own intelligence, because he has never been allowed to make the attempt. Statues and formulas, these mechanical tools of a serviceable use, or rather misuse, of his natural faculties, are the ankle-chains of a continuous immaturity. Whoever threw it off would make anuncertain jump over the smallest trench because he is not accustomed to such free movement. Therefore there are only a few who have pursued a firm path and have succeeded in escaping from immaturity by their own cultivation of the mind.But it is more nearly possible for a public to enlighten itself: this is even inescapable if only the public is given its freedom. For there will always be some people who think for themselves, even among, the self-appointed guardians of the great mass who, after having thrown off the yoke of immaturity themselves, will spread about them the spirit of a reasonable estimate of their own value and of the need for every man to think for himself.(43)______.Through revolution, the abandonment of personal despotism may be engendered and the end of profit-seeking and domineering oppression may occur, but never a true reform of the state of mind. Instead, new prejudices, just like the old ones, will serve as the guiding reins of the great, unthinking mass. (44)______.But I hear people clamor on all sides: Don't argue ! The officer says: Don't argue, drill! The tax collector: Don't argue, pay! The pastor: Don't argue, believe!... Here we have restrictions on freedom everywhere. Which restriction is hampering enlightenment, and which does not, or even promotes it? I answer; The public use of a man's reason must be free at all times, and this alone can bring enlightenment among men.The question may now be put: Do we live at present in an enlightened age'?(45)______.[A] All that is required for this enlightenment is freedom; and particularly the least harmful of that may be called freedom, namely, the freedom for man to make public use of his reason in all matters.[B] Enlightenment is man's leaving his self-caused immaturity. Immaturity is the incapacity to use one's intelligence without the guidance of another. Such immaturity is self-caused if it is not caused by lack of intelligence, but by lack of determination and courage to use one's intelligence without being guided by another. Have the courage to use your own intelligence! It is therefore the motto of the Enlightenment.[C] The answer is: No, but in an age of enlightenment. Much still prevents men from being placed in a position to use their own minds securely and well in matters of religion. But we do have very definite indications that this field of endeavor is being opened up for men to work freely and reduce gradually the hindrances preventing a general enlightenment and an escape from self-caused immaturity.[D] I call this soaring wealth and shrinking spirit "the American paradox." More than ever, we have big houses and broken homes, high incomes and low morale, secured rights and diminished civility. We excel at making a living but often fail at making a life. We celebrate our prosperity but yearn for purpose. We celebrate our freedoms but long for connection. In an age of plenty, we feel spiritual hunger.[E] A public can only arrive at enlightenment slowly.[F] After having made their domestic animals dumb and having carefully prevented these quiet creatures from daring to take any step beyond the lead-strings to which they have fastened them, these guardians then show them the danger which threatens them, should they attempt to walk alone. Now this danger is not really so very great; for they would presumably learn to walk after some stumbling. However, an example of this kind intimidates and frightens people out of allfurther attempts.[G] Attitudes about divorce have dramatically changed. In the past, it was seen as the last resort of a woman who had been beaten up or cheated on. Women were victims. Today many women choose' to get divorced because they think they will have a better life as a SINDI than by staying in a stale marriage. In 74 percent of eases it is women who now instigate divorce proceedings.41.42.43.44.45.Part CDirections:Read the following text carefully and then translate the underlined segments into Chinese. Your translation should be written clearly on ANSWER SHEET 2. (10 points)Almost all our major problems involve human behavior, and they cannot be solved by physical and biological technology alone. What is needed is a technology of behavior, but we have been slow to develop the science from which such a technology might be drawn, (46) One difficulty is that almost all of what is called behavioral science continues to trace behavior to states of mind, feelings, traits of character, human nature, and so on. Physics and biology once followed similar practices and advanced only when they discarded them, (47) The behavioral sciences have been slow to change partly because the explanatory items often seem to be directly observed and partly because other lands of explanations have been hard to find. The environment is obviously important, but its role has remained obscure. It does not push or pull, it selects, and this function is difficult to discover and analyze. (48). The role of natural selection in evolution was formulated only a little more than a hundred years ago, and the selective role of the environment in shaping and maintaining the behavior of the individual is only beginning to be recognized and studied. As the interaction between organism and environment **e to be understood, however, effects once assigned to states of mind, feelings, and traits are beginning to be traced to accessible conditions, and a technology of behavior may therefore become available. It will not solve our problems, however, until it replaces traditional pre-scientific views, and these are strongly entrenched. Freedom and dignity illustrate the difficulty. (49) They are the possessions of the autonomous (self-governing) man of traditional theory, and they are essential to practices in which a person is held responsible for his conduct and given credit for his achievements. A scientific analysis shifts both the responsibility and the achievement to the environment. It also raises questions concerning "values". Who will use a technology and to what ends? (50) Until these issues are resolved, a technology of behavior will continue to be rejected, and with impossibly the only way to solve our problems.。

高等数学单元测试题一(极限+连续)及答案

高等数学单元测试题一（极限+连续）一、选择题（每小题4分，共20分） 1、当0x →+时，（）无穷小量。

A.1sin x xB. 1x e C. ln x D. 1sin x x2、点1x =是函数311()1131x x f x x x x -<⎧⎪==⎨⎪->⎩的（）。

A.连续点B.第一类非可去间断点C.可去间断点D.第二类间断点3、函数()f x 在点0x 处有定义是其在0x 处极限存在的（）。

A.充分非必要条件 B.必要非充分条件 C.充要条件 D.无关条件4、已知极限22lim()0x x ax x→∞++=，则常数a 等于（）。

A.-1B.0C.1D.25、极限201lim cos 1x x e x →--等于（）。

A.∞B.2C.0D.-2 二、填空题（每小题4分，共20分）1、21lim(1)x x x→∞-=1、当0x →+时，无穷小ln(1)Ax α=+与无穷小sin 3x β=等价，则常数A=2、已知函数()f x 在点0x =处连续，且当0x ≠时，函数21()2x f x -=，则函数值(0)f =3、111lim[]1223(1)n n n →∞+++••+=14、若lim ()x f x π→存在，且sin ()2lim ()x xf x f x x ππ→=+-，则lim ()x f x π→= 三、解答题（8小题，共60分） 1、（7分）计算极限 222111lim(1)(1)(1)23n n→∞---2、（7分）计算极限 30tan sin lim x x xx→-3、（7分）计算极限 123lim()21x x x x +→∞++4、（7分）计算极限 01x e →-5、（7分）设3214lim 1x x ax x x →---++ 具有极限l ，求,a l 的值6、（8分）设3()32,()(1)n x x x x c x αβ=-+=-，试确定常数,c n ，使得()()x x αβ7、（7分）试确定常数a ，使得函数21sin 0()0x x f x xa x x ⎧>⎪=⎨⎪+≤⎩在(,)-∞+∞连续8、（10分）设函数()f x 在开区间(,)a b 连续，12a x x b <<<，试证：在开区间(,)a b 至少存在一点c ，使得11221212()()()()(0,0)t f x t f x t t f c t t +=+>>高等数学单元测试题一（极限+连续）参考答案一、选择题（每小题4分，共20分）1、A2、C3、D4、A5、D二、填空题（每小题4分，共20分）1、2e-2、 33、 04、 15、 1三、解答题1、（7分）解：原式=132411111 lim()()()lim223322 n nn n nn n n→∞→∞-++•••=•=2、（7分）解：原式=2322 000sin1sin1cos1 cos2lim lim limcos cos2 x x xxx xxxx x x x x→→→--===3、（7分）解：原式=11122112221lim(1)lim(1)121211lim(1)lim(1)1122xxx xxx xx xe x x+++→∞→∞+→∞→∞+=+++=+•+=++4、（7分）解：原式=201sin 12lim 2x x xx →=5、（7分）解：因为1lim(1)0x x →-+=，所以 321lim(4)0x x ax x →---+=，因此 4a = 并将其代入原式321144(1)(1)(4)lim lim 1011x x x x x x x x l x x →-→---++--===++ 6、（8分）解：32221()32(1)(2)(1)(2)3lim ,3,2(1)x x x x x x x x c n c x cα→=-+=-+-+=∴==- 此时，()()x x αβ7、（7分）解：当0x >时，()f x 连续，当0x <时，()f x 连续。

考研数学一(函数、极限与连续)历年真题试卷汇编1(题后含答案及解析)

考研数学一（函数、极限与连续）历年真题试卷汇编1(题后含答案及解析)题型有：1. 选择题 2. 填空题 3. 解答题选择题下列每题给出的四个选项中，只有一个选项符合题目要求。

1．(2003年)设{an}，{bn}，{cn}均为非负数列，且则必有( )A．an＜bn对任意n成立B．bn＜cn对任意n成立C．极限不存在D．极限不存在正确答案：D解析：由于则由极限的保号性可知，存在N＞0，使得当n＞N时，an＜bn，但不是对任意的n都成立。

例如bn=1，n=1，2时不满足an＜bn，所以选项A错误。

类似地，选项B也是错误的。

例如bn=1，n=1，2时不满足bn＜cn。

由于因此是0·∞型的未定式，有可能收敛也有可能发散，所以选项C是错误的。

例如极限证明发散，可采用反证法。

假设是收敛的，由于可知也是收敛的，与已知条件矛盾，假设不成立，也即是发散的。

由此唯一正确的选项是D。

知识模块：函数、极限与连续2．(2007年)设函数f(x)在(0，+∞)上具有二阶导数，且f”(x)＞0，令un=f(n)(n=1，2，…)，则下列结论正确的是( )A．若u1＞u2，则(un}必收敛B．若u1＞u2，则{un}必发散C．若u1＜u2，则{un}必收敛D．若u1＜u2，则{un}必发散正确答案：D解析：方法一：设f(x)=x2，则f(x)在(0，+∞)上具有二阶导数，且f”(x)＞0，u1＜u2，但{un}={n2}发散，排除C；设则f(x)在(0，+∞)上具有二阶导数，且f”(x)＞0，u1＞u2，但收敛．排除B；设f(x)=一lnx，则f(x)在(0，+∞)上具有二阶导数，且f”(x)＞0，u1＞u2，但{un}={一lnn}发散，排除A。

故应选D。

方法二：由拉格朗日中值定理，有un+1一un=f(n+1)一f(n)=f′(ξn)(n+1—n)=f′(ξn)，其中n＜ξn＜n+1(n=1，2，…)。

由f”(x)＞0知，f′(x)单调增加，故f′(ξ1)＜f′(ξ2)＜…＜f′(ξn)＜…，所以于是当u2一u1＞0时，有故选D。

近30年数学考研真题高数极限连续部分

当
时，用“
”表示比高阶的无穷小，则下列式子中错误的是：（）
(A)
(B)
(C)
(D)
【答案】：D
（2013302）
函数
的可去间断点的个数为：（）
(A)
(B)
(C)
(D)
【答案】：C
（2014201）
当
时，若
，
均是比高阶的无穷小，则的取值范围是：
(A)
(B)
(C)
(D)
【答案】：B
（2014301）
f
x 的间断点,其结论为
(
)
(A) 不存在间断点
(B) 存在间断点 x 1
(C) 存在间断点 x 0
(D) 存在间断点 x 1
【答案】：B
（1999209）
“对任意给定的 e 0,1，总存在正整数 N ，当 n N 时，恒有 | xn a | 2 ”是数列 xn
收敛于 a 的（）
（A）充分条件但非必要条件（C）充分必要条件【答案】：C
()
(C) 1 x2 1
【答案】：D
(D) x tan x
（1993202）
当
x
0时，变量
1 x2
sin
1 x
是则（
）
（A）无穷小
（C）有界的，但不是无穷小
【答案】：D
（B）无穷大（D）有界的，但不是无穷大
（1994102）
设 lim x0
a tan x b(1 cos x) c ln(1 2x) d (1 ex2
【答案】：B
（2008301）
设函数在区间
上连续，则
是函数
的：（）

高等数学函数的极限与连续习题精选和答案

1、函数()12++=x x x f 与函数()113--=x x x g 相同．错误 ∵当两个函数的定义域和函数关系相同时，则这两个函数是相同的。

∴()12++=x x x f 与()113--=x x x g 函数关系相同，但定义域不同，所以()x f 与()x g 是不同的函数。

2、如果()M x f >（M 为一个常数），则()x f 为无穷大．错误根据无穷大的定义，此题是错误的。

3、如果数列有界，则极限存在．错误如：数列()nn x 1-=是有界数列，但极限不存在4、a a n n =∞→lim ，a a n n =∞→lim ．错误如：数列()nn a 1-=，1)1(lim =-∞→nn ，但n n )1(lim -∞→不存在。

5、如果()A x f x =∞→lim ，则()α+=A x f （当∞→x 时，α为无穷小）．正确根据函数、极限值、无穷小量的关系，此题是正确的。

6、如果α～β，则()α=β-αo ．正确 ∵1lim=αβ，是 ∴01lim lim =⎪⎭⎫⎝⎛-=-αβαβα，即βα-是α的高阶无穷小量。

7、当0→x 时，x cos 1-与2x 是同阶无穷小．正确 ∵2122sin 412lim 2sin 2lim cos 1lim2022020=⎪⎪⎪⎪⎭⎫ ⎝⎛⋅⋅==-→→→x x x x x x x x x 8、 01sin lim lim 1sin lim 000=⋅=→→→xx x x x x x ．错误 ∵xx 1sin lim 0→不存在，∴不可利用两个函数乘积求极限的法则计算。

9、 e x xx =⎪⎭⎫⎝⎛+→11lim 0．错误 ∵e x xx =⎪⎭⎫⎝⎛+∞→11lim10、点0=x 是函数xxy =的无穷间断点．错误 =-→x x x 00lim 1lim 00-=--→x x x ，=+→x x x 00lim 1lim 00=+→xx x ∴点0=x 是函数xxy =的第一类间断点．11、函数()x f x1=必在闭区间[]b a ,取得最大值、最小值．错误 ∵根据连续函数在闭区间上的性质，()x f x1=在0=x 处不连续 ∴函数()x f x1=在闭区间[]b a ,不一定取得最大值、最小值二、填空题：1、设()x f y =的定义域是()1,0，则（１）()xef 的定义域是（ (,0)-∞ ）；（２）()x f 2sin 1-的定义域是（ ,()2x x k x k k Z πππ⎧⎫≠≠+∈⎨⎬⎩⎭）；（３）()x f lg 的定义域是（ (1,10) ）．答案：（1）∵10<<xe （2）∵1sin 102<-<x （3）∵1lg 0<<x2、函数()⎪⎩⎪⎨⎧≤<-=<<-+=403000222x x x x x x f 的定义域是（ (]4,2- ）．3、设()2sin x x f =，()12+=ϕx x ，则()[]=ϕx f （ ()221sin +x ）．4、nxn n sinlim ∞→＝（ x ）．∵x x n x n x n n x n x n n n n =⋅==∞→∞→∞→sinlim 1sin limsin lim 5、设()11cos 11211xx x f x x x x π-<-⎧⎪⎪=-≤≤⎨⎪->⎪⎩，则()10lim x f x →--=（ 2 ），()=+→x f x 01lim （ 0 ）． ∵()1010lim lim (1)2x x f x x →--→--=-=，()()01lim lim 0101=-=+→+→x x f x x6、设()⎪⎩⎪⎨⎧=≠-=00cos 12x ax x x x f ，如果()x f 在0=x 处连续，则=a （ 21 ）．∵21cos 1lim 20=-→x x x ，如果()x f 在0=x 处连续，则()a f xx x ===-→021cos 1lim 20 7、设0x 是初等函数()x f 定义区间的点，则()=→x f x x 0lim （ ()0x f ）．∵初等函数()x f 在定义区间连续，∴()=→x f x x 0lim ()0x f8、函数()211-=x y 当x →（ 1 ）时为无穷大，当x →（ ∞ ）时为无穷小．∵()∞=-→2111limx x ，()011lim2=-∞→x x9、若()01lim2=--+-+∞→b ax x x x ，则=a （ 1 ），=b （ 21-）． ∵()b ax x xx --+-+∞→1lim2()()()bax x x bax x x b ax x x x +++-+++---+-=+∞→111lim 222()()b ax x x b ax x x x +++-+-+-=+∞→11lim 222()()()b ax x x b x ab x a x +++--++--=+∞→11211lim 2222欲使上式成立，令012=-a ，∴1a =±，上式化简为()()()2211212112lim lim lim1x x x bab ab x b ab a →+∞→+∞→+∞--++-++--+==+∴1a =，021=+ab ，12b =-10、函数()x x f 111+=的间断点是（ 1,0-==x x ）． 11、()34222+--+=x x x x x f 的连续区间是（ ()()()+∞∞-,3,3,1,1, ）．12、若2sin 2lim =+∞→x xax x ，则=a （ 2 ）． ()200lim sin 2lim sin 2lim =+=+=⎪⎭⎫ ⎝⎛+=+∞→∞→∞→a a x x a x x ax x x x ∴2=a13、=∞→x x x sin lim（ 0 ），=∞→xx x 1sin lim （ 1 ）， ()=-→xx x 11lim （ 1-e ），=⎪⎭⎫ ⎝⎛+∞→kxx x 11lim （ ke ）． ∵0sin 1lim sin lim=⋅=∞→∞→x x xx x x 111sin lim1sin lim ==∞→∞→xx x x x x()[]1)1(110)(1lim 1lim --⋅-→→=-+=-e x x xx x x k kx x kxx e x x =⎥⎦⎤⎢⎣⎡+=⎪⎭⎫ ⎝⎛+∞→∞→)11(lim 11lim14、limsin(arctan )x x →∞=（不存在），lim sin(arccot )x x →+∞=（ 0 ）三、选择填空：1、如果a x n n =∞→lim ，则数列n x 是（ b ）a.单调递增数列 b ．有界数列 c ．发散数列2、函数()()1log 2++=x x x f a 是（ a ）a ．奇函数b ．偶函数c ．非奇非偶函数 ∵()()11log 1)(log 22++=+-+-=-x x x x x f aa()()x f x x a -=++-=1log 23、当0→x 时，1-xe 是x 的（ c ）a ．高阶无穷小b ．低阶无穷小c ．等价无穷小4、如果函数()x f 在0x 点的某个邻域恒有()M x f ≤（M 是正数），则函数()x f 在该邻域（ c ）a ．极限存在b ．连续c ．有界5、函数()x f x-=11在（ c ）条件下趋于∞+. a ．1→x b ．01+→x c ．01-→x6、设函数()x f xxsin =，则()=→x f x 0lim （ c ）a ．１b ．－１c ．不存在 ∵1sin lim sin limsin lim000000-=-=-=-→-→-→xx x x x xx x x1sin lim sin lim 0000==-→+→xx x x x x 根据极限存在定理知：()x f x 0lim →不存在。

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考研数学训练题
高等数学1——极限与连续练习题
1．填空题
（1）极限=+-+∞
→)]1ln()3[ln(lim x x x x ；（ 2 ）（2）已知极限82lim =⎪⎭
⎫ ⎝⎛-+∞→x x a x a x ，则=a ；（ 2ln ）（3）已知极限)0()1(lim 2008
≠=--∞→A A n n n k k n ，则=k ，=A ；（2009，2009
1）（4）已知当0→x 时，
1)1(3/12-+ax 与1cos -x 是等价无穷小，则=a ；（ 23- ）（5）若极限0)(6sin lim
30=+→x x xf x x ，则=+→20)(6lim x x f x ；（ 36 ）（6）若极限2)2(lim 0
=→x f x x ，则极限=→x x f x )4(lim 0 ；（ 1 ）（7）设)(x p 是多项式，且2)(lim 23=-∞→x x x p x ，1)(lim 0=→x
x p x ，则=)(x p ；（ x x x x p ++=2
32)( ）
（8）曲线1e 1
+=x x y 的斜渐近线是；（ 2+=x y ）（9）当0≠x 时，函数232e 2cos )(2x
x x f x -+=，则=)0(f 时，函数)(x f 在)(∞+-∞，内连续；（ 1 ）
（10）设函数bx
a x x f e )(+=在)(∞+-∞，内连续，且0)(lim =-∞→x f x ，则常数
b a 、应满足．（ 00≥<a b ，）
2．单项选择题
（1）如果极限6)31)(21)(1(lim 0=++++→x
a x x x x ，则=a （）；（A ）（A ） 1-；（B ） 1；（C ） 2；（D ） 3．
（2）若极限2)e 1()21ln()
cos 1(tan lim 20=-+--+-→x x d x c x b x a ，其中022≠+c a ，则必有（）（D ）
（A ）d b 4=；（B ）d b 4-=；（C ）c a 4=；（D ）c a 4-=．
（3）当1→x 时，函数112e 1
1)(---=x x x x f 的极限（）；（D ）（A ）等于2；（B ）等于0；（C ）是∞；（D ）不存在，但不是∞．
（4）设函数232)(-+=x x x f ，当0→x 时，)(x f 是x 的（）无穷小；（B ）
（A ）等价；（B ）同阶但不等价；（C ）高阶；（D ）低阶．
（5）设函数43sin 02)(,d sin )(x x x g t t x f x
+==⎰，
当0→x 时，)(x f 是)(x g 的（）无穷小；（B ）
（A ）等价；（B ）同阶但不等价；（C ）高阶；（D ）低阶．
（6）当0→x 时，函数)1(e )(2
++-=bx ax x f x 是比2x 的高阶无穷小，则（）；（A ）（A ）121==b a ，；（B ）11==b a ，；（C ）12
1=-=b a ，；（D ）11=-=b a ，．（7）当0→x 时，函数x x x f e e )(tan -=与k ax 为等价无穷小，则（）；（C ）
（A ）131
==k a ，；（B ）313==k a ，；（C ）331==k a ，；（D ）3
131==k a ，．（8）若数列}{}{n n y x 、满足0lim =∞
→n n n y x ，则下列断言中正确的是（）；（D ）（A ）若数列}{n x 发散，则数列}{n y 也发散；（B ）若数列}{n x 无界，则数列}{n y 必有界；
（C ）若数列}{n x 有界，则n y 为必无穷小；（D ）若n
x 1为无穷小，则n y 为必为无穷小．（9）设函数)()(x g x f 、在)(∞+-∞，内有定义，0)(≠x f 为连续函数，)(x g 有间断点，则（）必有间断点；（D ）
（A ）)]([x f g ；（B ） 2)]([x g ；（C ） )]([x g f ；（D ） )
()(x f x g ．（10）0=x 点是函数x
x f 1arctan )(=的（）间断点. (A) (A) 可去； (B) 跳跃； (C) 无穷型； (D) 振荡型.
3．求极限1lim 21--+++→x n x x x n x Λ （n 是正整数）. （ 2
)1(+n n ） 4．求极限)1
arctan (arctan lim 2+-∞→n a n a n n . （ a ）
5．求极限⎪⎪⎭
⎫ ⎝⎛+++→x x x x x sin e 1e 2lim /4/10. （ 1 ） 6．求极限)]11ln([lim 2x x x x +-+∞→. （ 2
1 ） 7．求极限∑=∞→++n k n k k k 1)2)(1(1lim
. ( 41 ） 8．求极限∑=∞→n k n n n k 1lim . （ 3
2） 9．求极限131)
1()1()1)(1(lim -→----n n x x x x x Λ. （ !1n ） 10．求极限∏=∞→n k k n x 12cos lim （0≠x ）. （ x
x sin ） 11．求极限n n n n n n )12()1(1lim -+∞→Λ. （ e
4 ） 12．已知极限0arctan 1)(1lim 220
≠=-+→c x x x f x ，求b a 、的值，使得0→x 时，)(x f 与b ax 为等价无穷小. （ 42==b c a 、，34P 例48 ）
13．已知极限A a x x f x x =-+
→1
)sin )(1ln(lim 0（10≠>a a ，），求极限20)(lim x x f x →.（a A ln ，36P 例53 ） 14．已知极限0)3sin (lim 230=++→b x a x x x ，求常数b a 、．（ 293=-=b a ，） 15．已知211
)(x x f +='，x x g +='11)(，且0)0()0(==g f ，求极限⎥⎦
⎤⎢⎣⎡-→)(1)(1lim 0x g x f x . （ 2/1- ）
16．设)(x f 是三次多项式，且满足14)(lim 2)(lim 42=-=-→→a
x x f a x x f a x a x （0≠a ），求极限a x x f a x 3)(lim 3-→. （ 2
1- ） 17．设函数⎪⎩
⎪⎨⎧=-≠≠-+++=.1,2,2,1,2)(22x x x x x b ax x x f 求a 、b 的值，使得函数)(x f 在1=x 点
连续. （ 4=a 、5-=b ）
18．设函数x
x b a x x x f 22sin sin sin sin 1)(--++=，若0=x 是)(x f 的可去间断点，求a 、b 的值，并求)(lim 0x f x →. （ 211==b a ，，83)(lim 0=→x f x ）。