Coordination of aproximations in secondary school students’ understanding of limit concept

The aim of this research is to characterize the coordination of the processes of approximation related to the understanding of the limit of a function. We analyze the answers of 64 post-secondary school students to 7 problems considering the dynamic and metric conception of limit of a function. Results indicate that the metric understanding of the limit in terms of inequality supports that the student is capable of coordinating the approximations in the domain and in the range when lateral approximations coincide. However, the student is not capable of this coordination when lateral approximations do not coincide. This indicates that the metric understanding of the limit begins with the previous construction of the dynamic conception in case of coincidence of the lateral approximations in the range.

An Acyl-NHC Osmium Cooperative System: Coordination of Small Molecules and Heterolytic B–H and O–H Bond Activation

The hexahydride complex OsH6(PiPr3)2 (1) activates the C–OMe bond of 1-(2-methoxy-2-oxoethyl)-3-methylimidazolium chloride (2), in addition to promoting the direct metalation of the imidazolium group, to afford a five-coordinate OsCl(acyl-NHC)(PiPr3)2 (3) compound. The latter coordinates carbon monoxide, oxygen, and molecular hydrogen to give the corresponding carbonyl (4), dioxygen (5), and dihydrogen (6) derivatives. Complex 3 also promotes the heterolytic bond activation of pinacolborane (HBpin), using the acyl oxygen atom as a pendant Lewis base. The hydride ligand and the Bpin substituent of the Fischer-type carbene of the resulting complex 7 activate the O–H bond of alcohols and water. As a consequence, complex 3 is a metal ligand cooperating catalyst for the generation of molecular hydrogen, by means of both the alcoholysis and hydrolysis of pinacolborane, via the intermediates 7 and 6.