共查询到19条相似文献,搜索用时 265 毫秒
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等价无穷小代换在计算极限中是一种非常有用的方法.特别对幂指函数极限的计算,如能巧妙运用,可使问题变得简明、易解,为此我们介绍以下命题.命题设在自变量的某一变化过程中,均为无穷小量,若且证明由例1求极限解当在以上计算中,我们采用了等价无穷小代换,使问题变得简明、易解,如利用罗必塔法则或其他方法将是很繁锁的,读者不妨一试.有些幂指函数极限,还可用等价无穷小代换并结合重要权限及导数定义等综合求解.例4设存在,试求当时故原式幕指函数极限的一种简捷求法@符世斌$陕西财经学院!西安,710061 相似文献
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学生用等价无穷小代换求极限的过程中出现许多我们认为是错误的方法,但有些错误的方法又能够得到正确的结论,如求极限时对复合函数的中间变量作等价无穷小代换;两非等价无穷小和差分别做等价无穷小代换等,这应该算对还是错?本文在理论上作了分析. 相似文献
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<正> 文[1]论述了等价无穷小代换在求极限中的重要作用。本文进一步强调和说明等价无穷小的思想贯穿于《高等数学》的许多重要的基本概念和基本方法中,在教学过程中应用等价无穷小的思想去理解这些概念和方法,无疑对整个高等数学概念的理解是有益的。 相似文献
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在求函数极限的过程中,利用等价无穷小代换常常会使计算简单,但对形如的极限,若无穷小量有时权限并不相等.这里的关键是与的选取不当.本文着重讨论这种情况下利用Taylor公式选取适当的等价无穷小量作代换保持权限不变.为叙述方便,我们只讨论的情况.同时假定下文中所涉及的都是当时的连续且具有任意阶导数的函数无穷小量,以后不再交代.引理1若引理2若则,这里不全为0引理3若a在处的Taylor展开式(带皮亚诺余项)为(按前面假设,常数项为0):证明显然,从而实际上,通常我们所用的等价无穷小都是取Tayfor展式的第一个非零项.如… 相似文献
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普遍认为利用等价无穷小代换求代数和及复合函数的极限时常常隐藏着两个误区,通过对其进行探讨可以发现.只要加以适当的条件,代数和各部分为无穷小量以及复合函数的中间变量为无穷小量时.对这些部分是可以进行等价无穷小代换的. 相似文献
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一、利用等价无穷小代换来求极限的一些容易证明的定理定理1设无穷小量f(x)~(x),且limf(x)·g(x)存在,则这里,(1)无穷小量f(x)~(x),表示f(x)与(x)是当x→x0或x→∞时的等价无穷小;(2)limf(x)表示limf(x)或limf(x).下同。定理2设无穷小量f(x)~(x),且存在,则由这二个定理可知,一般在乘或除的情况下是可用等价无穷小代换来求极限的。此外在幂指函数求极限中,也常利用等价无穷小代换,这有下面二个定理,这里只证后一个定理。定理3设八x)>0,无穷小量g(x)~~(x),且tim八x)”“’存在,则定… 相似文献
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<正> 无穷小的比较及等价无穷小的一些概念和性质,在极限运算,数项级数及广义积分等教学环节中恰当的运用,对提高学生运算能力,进一步认识这些问题的实质起着不可忽视的作用。 相似文献
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In this paper we derive general upper bounds for the total variation distance between the distributions of a partial sum process in row-wise independent, non-negative triangular arrays and the sum of a fixed number of corresponding extremal processes. As a special case we receive bounds for the supremum distance between the distribution functions of a partial sum and the sum of corresponding upper extremes which improve upon existing results. The outcome may be interpreted as the influence of large insurance claims on the total loss. Moreover, under an additional infinitesimal condition we also prove explicit bounds for limits of the above quantities. Thereby we give a didactic and elementary proof of the Ferguson–Klass representation of Lévy processes on ??≥?0 which reflects the influence of extremal processes in insurance. 相似文献
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《Journal of Pure and Applied Algebra》2022,226(5):106925
In this article we give a new proof for the infinitesimal Torelli theorem for minimal elliptic surfaces without multiple fibers with Euler number at least 24 for nonconstant j-invariant. In the case of constant j-invariant we find a new proof in the case of Euler number at least 72. We also discuss several new counterexamples. 相似文献
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基于经典分析数学的观点,针对黄乘规新著《标准的欧弥伽无穷小微积分学》一书中引入"无穷小线段"等问题,提出一些需要探讨和再思的问题,以供高校数学分析课程教学的师生们参议评论.同时指出欧弥伽微积分缺乏计算功能的原因,供参考研究. 相似文献
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Jun Hu 《Proceedings of the American Mathematical Society》2005,133(1):193-202
The infinitesimal earthquake theorem gives a one-to-one correspondence between Thurston bounded earthquake measures and normalized Zygmund bounded functions. In this paper, we provide an intrinsic proof of a theorem given in an earlier paper by the author; that is, we show that the cross-ratio norm of a Zygmund bounded function is equivalent to the Thurston norm of the earthquake measure in the correspondence.
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History and research on proof by contradiction suggests proof by contradiction is difficult for students in a number of ways. Students’ comprehension of already-written proofs by contradiction is one such aspect that has received relatively little attention. Applying the cognitive lens of Action-Process-Object-Schema (APOS) Theory to proof by contradiction, we constructed and tested a cognitive model that describes how a student might construct the concept ‘proof by contradiction’ in an introduction to proof course. Data for this study was collected from students in a series of five teaching interventions focused on proof by contradiction. This paper will report on two participants as case studies to illustrate that our cognitive trajectory for proof by contradiction is a useful model for describing how students may come to understand the proof method. 相似文献
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Motivated by comatrix coalgebras, we introduce the concept of a Newtonian comatrix coalgebra. We construct an infinitesimal unitary bialgebra on matrix algebras, via the construction of a suitable coproduct. As a consequence, a Newtonian comatrix coalgebra is established. Furthermore, an infinitesimal unitary Hopf algebra, under the view of Aguiar, is constructed on matrix algebras. By the close relationship between pre-Lie algebras and infinitesimal unitary bialgebras, we erect a pre-Lie algebra and a new Lie algebra on matrix algebras. Finally, a weighted infinitesimal unitary bialgebra on non-commutative polynomial algebras is also given. 相似文献
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V. I. Arnold proved in 1993 that the intersection multiplicity between two germs of analytic subvarieties at a fixed point of a holomorphic invertible self-map remains bounded when one of the germs is dragged by iterations of the self-map. The proof is based on the Skolem-Mahler-Lech theorem on zeros in recurrent sequences. We give a different proof, based on the Noetherianity of certain algebras, which allows one to generalize Arnold’s theorem for local actions of arbitrary finitely generated commutative groups, with both discrete and infinitesimal generators. Simple examples show that for non-commutative groups the analogous assertion fails. 相似文献
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Journal of Algebraic Combinatorics - The concept of weighted infinitesimal unitary bialgebra is an algebraic meaning of the nonhomogenous associative Yang–Baxter equation. In this paper, we... 相似文献