【正文】 ing) 設(shè), 是兩個(gè)非空集合 , 如果存在一個(gè)法則， 使得對中每個(gè)元素， 按法則， 在中有唯一確定的元素與之對應(yīng) , 則稱為從到 的映射 , 記作。Let , be two nonempty sets, if there exists a rulethat associates a unique element of to every element in , then is called a mapping from to ，denoted by .三、函數(shù) (Function)函數(shù)概念 (The Concept of Function) 設(shè)數(shù)集, 則稱映射為定義在上的函數(shù) , 通常簡記為, 其中 稱為自變量, 稱為因變量,稱為定義域 , 記作。Let the number set ， then the mapping is called a function defined on ， usually denoted by ， where is called an independent variable, is called a dependent variable, is called a domain, denoted by . 數(shù)列的極限 (Limit of the Sequence of Number)一、數(shù)列極限的定義 (Definition of the Limit of Sequence of Number)設(shè)為一一數(shù)列 , 如果存在常數(shù)，對于任意給定的正數(shù) ( 不論它多么小 ) , 總存在正整數(shù)，使得當(dāng)時(shí) ,不等式都成立 , 那么就稱常數(shù)是數(shù)列的極限 , 或者稱數(shù)列收斂于，記為 或 。Let be a sequence of number, If there exists a constant , such that for any given positive number , there exists a positive integer , such that for every , ,then the constant is called the limit of the sequence ,or we call that the sequence converges to , denoted by or 二、收斂數(shù)列的性質(zhì) (Properties of Convergent Sequence)定理 1 ( 極限的唯一性 ) 如果數(shù)列收斂 , 那么它的極限唯一。Theorem 1 (Uniqueness of Limit) If the sequence is convergent, then its limit is unique. 定理 2 ( 收斂數(shù)列的有界性 ) 如果數(shù)列收斂 , 那么數(shù)列必定有界。Theorem 2 (Boundedness of a Convergent Sequence) If the sequenceis convergent, thenis bounded.定理3 如果,且(或),那么存在正整數(shù)，當(dāng)時(shí)，都有(或)。Theorem 3 If ,and (or ),then there exists a positive integer ,such that (或),for every .定理4(收斂數(shù)列與其子數(shù)列間的關(guān)系) 如果數(shù)列收斂于，那么它的任一子數(shù)列也收斂，且極限也是 。Theorem 4(Relation of a Convergent Sequence between its Subsequence) If the sequence converges to ,then any of its subsequence is also convergent, and the limit is also . 函數(shù)的極限 (Limit of Function) 一、函數(shù)極限的定義 (Definition of Limit of Function) 1. 自變量趨于有限值時(shí)函數(shù)的極限 (Definition of Limit of Function (as Tends to a Real Number )).定義 1設(shè)函數(shù)在某個(gè)包含的鄰域內(nèi)有定義(可能在無定義)，如果存在常數(shù)，對于任意給定的正數(shù)，總存在正數(shù)，使得當(dāng)滿足不等式時(shí)，對應(yīng)的函數(shù)值都滿足不等式,那么常數(shù)就叫做函數(shù)當(dāng)時(shí)的極限 , 記作或(當(dāng))。Definition 1 Let be a function defined on some neighborhood of ，maybe not at itself. If there exists a constant， such that for any given positive number , there exists a positive number ， such that if ，then. Then the constant is called the limit ofas approaches .Denoted by or(as).2. 自變量趨于無窮大時(shí)函數(shù)的極限 (Definition of Limit of Function (as Tends to Infinity))定義 2 設(shè)函數(shù)當(dāng)大于某一正數(shù)時(shí)有定義。如果存在常數(shù)， 對于任意給定的正數(shù), 總存在正數(shù), 使得當(dāng)滿足不等式時(shí)，對應(yīng)的函數(shù)值都滿足不等式, 那么常數(shù)就叫做函數(shù)當(dāng) 時(shí)的極限 , 記作或（當(dāng)）。Definition 2 Let be a function which is defined if is greater than some positive number. If there exists a constant ，such that for any given positive number , there is a positive number ，such that if ，then .Then the constant is called the limit of as approaches . Denoted by or （當(dāng)）.二、函數(shù)極限的性質(zhì) (Properties of the Limit of Function) 定理 1( 函數(shù)極限的唯一性 ) 如果存在 , 那么這極限唯一。 Theorem 1 (Uniqueness of Limit of Function) If exists, then its limit is unique.定理 2( 函數(shù)極限的局部有界性 ) 如果,那么存在常數(shù),使得當(dāng)時(shí) , 有.Theorem2 (Locally Boundedness of the Limit of Function) If, then there exists ,such that implies . 定理 3 如果且(或)，那么存在常數(shù)，使得當(dāng)時(shí) , 有( 或) 。 Theorem 3 If，and (or)， then there exists a constant ，such that whenever then (or).定理 4 ( 函數(shù)極限與數(shù)列極限間的關(guān)系 ) 如果極限存在,為函數(shù)的定義域內(nèi)任一收斂于的數(shù)列 , 且滿足（）, 那么相應(yīng)的函數(shù)值列必收斂 , 且 。Theorem 4 (Relations between Limit of Function and Limit of Sequence) If exists, is any sequence in the domain of that converges to ,and satisfies （）,then the corresponding sequence of function value must be convergent, and . 無窮小與無窮大 (Infinitesimal and Infinity) 定義 1 如果函數(shù)當(dāng)( 或) 時(shí)的極限為零 , 那么稱函數(shù)為當(dāng)或（) 時(shí)的無窮小。 Definition 1 If the limit of function is zero as (or ), then is called an infinitesimal as (or ).定義 2 設(shè)函數(shù)在的某一去心鄰域內(nèi)有定義 ( 或大于某一正數(shù)時(shí)有定義 ) ,如果對于任意給定的正數(shù)，總存在正數(shù)( 或正數(shù)), 只要適合不等式( 或), 對應(yīng)的函數(shù)值總滿足不等式，則稱函數(shù)為當(dāng)（或 時(shí)的無窮大。 Definition 2 Let be a function defined on some neighborhood of ，maybe not at itself (or is defined for greater than some positive number), if for any given ，there exists (or)，such that whenever (or),thenis called an infinity as (or. 極限運(yùn)算法則 (Operation Rule of Limit) 定理 1 有限個(gè)無窮小的和也是無窮小。Theorem 1 The sum of finite number of infinitesimal is an infinitesimal.定理 2 有界函數(shù)與無窮小的乘積是無窮小。Theorem 2 The product of a bounded function and an infinitesimal is an infinitesimal.推論 1 常數(shù)與無窮小的乘積是無窮小。Corollary 1 The product of a constant and an infinitesimal is an infinitesimal.推論 2 有限個(gè)無窮小的乘積是無窮小。Corollary 2 The product of finite number of infinitesimal is an infinitesimal.定理 6( 復(fù)合函數(shù)的極限運(yùn)算法則 ) 設(shè)函數(shù)是由函數(shù)與函數(shù)復(fù)合而成,在點(diǎn)的某去心鄰域內(nèi)有定義 , 若，且存在，當(dāng)時(shí) , 有，則 Theorem 6 (Operation Rule of Limits of Composite Functions) Suppose the functionis the position of and ，and is defined on some neighborhood of (except possibly at ). If , and there exists, for , we have， then 極限存在準(zhǔn)則兩個(gè)重要極限 (Rule for the Existence of Limits Two Important Limits)準(zhǔn)則 I 如果數(shù)列、及滿足下列條件 :(1)（）, (2)，那么數(shù)列的極限存在 , 且 。Rule I Suppose the sequences 、and satisfy: (1) （）,(2) ，Then the limit of the sequence exists, and .準(zhǔn)則Ⅰ＇如果(1) 當(dāng) ( 或) 時(shí) ,(2) ，那么存在 , 且等于。Rule Ⅰ＇ Suppose(1) for (or), (2) ，then exists, and equals .準(zhǔn)則 II 單調(diào)有界數(shù)列必有極限。Rule II Every bounded monotonic sequence has limit. 準(zhǔn)則II＇設(shè)函數(shù)在點(diǎn)的某個(gè)左鄰域內(nèi)單調(diào)并且有界 ,則在的左極限必定存在。Rule II＇ Suppose the function is bounded monotonic on some left neighborhood of , then the lefthand limit of exists. 無窮小的比較 (The Comparison of Infinitesimal)定義如果,就說是比高階的無窮小 , 記作；如果, 就說是比低階的無窮小；如果, 就說與是同階的無窮?。蝗绻? ，就說是關(guān)于的的階的無窮?。蝗绻?，就說與是等階無窮小。記作；DefinitionIf ,then is called a higher order infinitesimal of ,denoted by。If ,then is called a lower order infinitesimal of .If ,then is called an infinitesimal of the same order as .If ,then is called an infinitesimal of the th order as .If ,then is called an equivalent infinitesimal of , denoted by 。定理1 與是等階無窮小的充分必要條件為 .Theorem 1 and are equivalent infinitesimals if and only if .定理2 設(shè)，且存在，則。 Theorem 2 Let ，and exists, then . 函數(shù)的連續(xù)性與間斷點(diǎn) (Continuity of Function and Discontinuity Points)函數(shù)的連續(xù)性 (Continuity of Func