Explain whylim(x,y)→(0,0)(3xy2x2+y4)doesn't exist. Hint! For one path considerx=y2. Limits of functions of two variables: We say that the functionf(x,y)approaches the LimitLas(x,y)approaches(x0,y0), and...
Thus you can construct an "infinite-tuple" x=(2,7,57,⋯) in the infinite direct product ∏Z/5i+1Z such that each elt xi+1 can be "mapped back" to xi using homomorphisms (in this case, modulo 5i+1). This construction precisely gives us a ring, and out "infinite tuple" ...
I wonder if there is such a mathematical expression which follows the definition of "multiplication" as in advanced calculus which actually provides the results on the table; i.e, some sort of bijective homomorphism that maps v: V*V --> V ...