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Applied Analysis
Published in Nirdosh Bhatnagar, Introduction to Wavelet Transforms, 2020
Thus f(x) has a limit L at x=x˜, if for the numbers x near x˜, the value of f(x) is close to L. The right-hand and left-hand limits are generally called the one-sided limits, and limx→x˜f(x) is called the two-sided limit. These three limits are related by the following lemma.
Scalars, vectors and fields
Published in Alan Jeffrey, Mathematics, 2004
The name ‘vector function’ suggests, correctly, that it is possible to give satisfactory meaning to the terms limit, continuity and derivative when applied to r(t). As in the ordinary calculus, the key concept is that of a limit. Intuitively the idea of a limit is clear: when we say u(t) tends to a limit v as t→ t0, we mean that when t is close to t0, the vector function u(t) is in some sense close to the vector v. In what sense, though, can the two vectors u(t) and v be said to be close to one another? Ultimately, all that is necessary is to interpret this as meaning that |u(t) − v| is small.
Single Degree-of-Freedom Undamped Vibration
Published in Haym Benaroya, Mark Nagurka, Seon Han, Mechanical Vibration, 2017
Haym Benaroya, Mark Nagurka, Seon Han
He is best known for developing the rule which bears his name for finding the limit of a rational function whose numerator and denominator tend to zero at a point. L’Hôpital’s rule uses derivatives to help evaluate limits involving indeterminate forms. Application (or repeated application) of the rule often converts an indeterminate form to a determinate form, allowing easy evaluation of the limit. L’Hôpital’s rule first appeared in his book.
Toward adaptive infrastructure: flexibility and agility in a non-stationarity age
Published in Sustainable and Resilient Infrastructure, 2019
Mikhail V. Chester, Braden Allenby
We characterize the confluence of constraints (barriers) that prevent us from achieving more desirable state as a limit. In mathematics, limits are values that functions approach as the inputs approach some value. In the case of infrastructure, limits can be thought of as the practical achievable futures given technical and non-technical barriers. The concept of a limit is important because it recognizes the reality that the infrastructure that we’ve integrated into all facets of society may not allow us to reach a desirable future, that there’s only so much better we can make it or the activities that use it. For example, given the preference and incentivizes for automobile technology focused roadway infrastructure there may not be a pathway where greenhouse gas emissions can be reduced fast enough to avoid some significant climate change threshold. Or consider that the design and deployment of housing stock (specifically the materials, building technologies, and form) may limit our ability to reduce building energy use beyond a certain threshold (Nahlik & Chester, 2015). As such, our current infrastructure and the forces that maintain their persistence are limiting our ability to achieve desired goals. More subtly, so is our educational system. Educating engineers to think of systems only in terms of traditional technological frameworks, such as energy, transportation, or information, makes it difficult to achieve integrated design and management of infrastructure, and thus impedes ability to reduce the deleterious effects of lock-in. And yet engineering education has its own path dependency, reflecting such constraints as professional and school accreditation (e.g. meeting accreditation requirements).
Pre-service student teachers’ conceptions of the notion of limit
Published in International Journal of Mathematical Education in Science and Technology, 2022
Sarah Bansilal, Thokozani Wiseman Mkhwanazi
The concept of the limit presents challenges to many mathematics students when encountered for the first time during their calculus courses. The concept of limit is a foundational concept for several concepts such as continuity, differentiation and integration. Research suggests that many student struggle with limits (Bezuidenhout, 2001; Cappetta & Zollman, 2009; Denbel, 2014; Limits, 1991; McCombs, 2014; Monaghan, 1991; Monaghan et al., 1994; Patel, 2013; Roh, 2010; Zollman, 2014). Although there is much research in Southern Africa focusing on mathematics education, studies about students’ understanding of concepts encountered in calculus such as derivatives, integration, limits and continuity are very poorly represented in the mathematics education research in the area. Since calculus forms a foundational part of the teacher education courses for prospective mathematics teachers, it is important that we engage in research about students’ difficulties with engaging with such concepts. This study is located within a Bachelor of Education (B.Ed) programme which takes the form of a four-year degree including courses focused on the mathematics specializations, pedagogic content knowledge and others which focus on developing pedagogic skills and overall disciplinary knowledge in education. As part of the mathematics specialization students who want to teach high school mathematics take on three courses in calculus. The participants in this study were enrolled in the initial differential calculus module. This is an exploratory study, which is underpinned by the following research question: What are the various meanings given by students to the concept of a limit? The findings of the study is intended to provide direction to future offerings of the course and to also add more insight to the research field about the language used by students to describe their understanding of advanced mathematical concepts. Furthermore, the findings provide support for findings in other studies in this area.