3138. Limits Evaluate the following limits or determine that they do not exist. lim 1x,y2S14,-22 110x - 5y + 6xy2

(Midterm 2) Theory Lemmas, theorems and definitions Polynomials over a ring R Definition: Polynomials over a ring R (with coefficients in R) are expressions of type r0 1+r2+........n where, x is referred to an indeterminate (n≥0 an1 2…..,nare coefficients) subject to certain conventions. Lemma: All polynomials together with ‘+’, ‘.’ forms a ring called polynomial ring over R; denoted as R[x] (contains R) Lemma: Suppose R is an integral domain . Let f(x), g(x) ∈ R[x]. Then deg(f(x).g(x))=deg f(x) + deg g(x) Theorem: Suppose R is an integral domain, then the ring R[x] is an integral domain. Theorem: Let R be an integral domain. Let f(x)∈R[x] Then f(x) is a unit in R[x] Division algorithm for polynomials Let F be a field Let a(x), b(x) ∈ F[x], then there exists polynomials q(x), r(x) satisfying 1. a(x)=b(x)q(x)+r(x) 2. r(x)=0 or deg r(x)