Corollary 9.1.2 Let f and g be defined on domains in V, a normed vector space,...

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Corollary 9.1.2 Let f and g be defined on domains in V, a normed vector space, with values in E, as in the Remark above. Suppose a is a cluster point of the intersection Den Dg of the domains off and g, lim, f(x) L and lime a g(x) - M. Then i. limsa (ft g)(x) = L + M. In ii. If m= 1, then lima(fg)(r) LM. iii. If m= 1. then limx(x) = provided that M 40 and that a is a cluster point of DL. Proof: The proofs are very similar to those of Corollary 2.1.1. Here we will treat only conclusion (i). Consider a sequence of points x,, E Den Dg\ {a} such that X a. Then (ft g)(x) = f(x₂) ± g(x₂)→ LM, by the corresponding theorem for limits of sequences. The other proofs are similar. This corollary suggests that limits of functions on E" are very similar in behavior to limits of real-valued functions of one real variable, subject only to certain essential hypotheses to insure the requisite operations are defined for the objects in question. However, the reader must be very careful: it is much harder for the limit of even a real-valued function of two or more variables to exist than it is for functions of only one variable. See Exercise 9.4. Definition 9.1.2 Let x EE" and let k = (k₁,..., kn) be any n-tuple of nonnegative integers. We define .... q a real-valued monomial on E". A real-valued polynomial on En is any function of the form p(x) = -1 Cjxk, where each c; R. A rational function is any ratio of two real-valued polynomials. A rational function is defined wherever the denominator is nonzero. Corollary 9.1.2 Let f and g be defined on domains in V, a normed vector space, with values in E, as in the Remark above. Suppose a is a cluster point of the intersection Den Dg of the domains off and g, lim, f(x) L and lime a g(x) - M. Then i. limsa (ft g)(x) = L + M. In ii. If m= 1, then lima(fg)(r) LM. iii. If m= 1. then limx(x) = provided that M 40 and that a is a cluster point of DL. Proof: The proofs are very similar to those of Corollary 2.1.1. Here we will treat only conclusion (i). Consider a sequence of points x,, E Den Dg\ {a} such that X a. Then (ft g)(x) = f(x₂) ± g(x₂)→ LM, by the corresponding theorem for limits of sequences. The other proofs are similar. This corollary suggests that limits of functions on E" are very similar in behavior to limits of real-valued functions of one real variable, subject only to certain essential hypotheses to insure the requisite operations are defined for the objects in question. However, the reader must be very careful: it is much harder for the limit of even a real-valued function of two or more variables to exist than it is for functions of only one variable. See Exercise 9.4. Definition 9.1.2 Let x EE" and let k = (k₁,..., kn) be any n-tuple of nonnegative integers. We define .... q a real-valued monomial on E". A real-valued polynomial on En is any function of the form p(x) = -1 Cjxk, where each c; R. A rational function is any ratio of two real-valued polynomials. A rational function is defined wherever the denominator is nonzero.

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Corollary 912 is a statement about the behavior of limits of functions in a normed vector space It s... View the full answer