1873_solutions

Is a nonnegative integer and r n then the set has

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Unformatted text preview: for every x then we ought to conclude that Ý  Ý doesn’t exist. If we define f x  x 2 and g x  x for all x then we ought to conclude that Ý  Ý ought to be Ý. It is clear that no single definition of the symbol Ý  Ý would be useful to us. 2. Thinking of the rule for products of limits lim f x g x xa  lim f x xa 91 lxim g x a that you saw in elementary calculus, give some examples to show why the expression Ý defined. If we define f x  x and g x  1 for all x  0 then x lim f x  Ý xÝ 0 should not be and lim g x  0 xÝ and lim f x g x  1. xÝ This example suggests that if Ý 0 ought to be defined then its value should be zero. However, if we were to define f x  x 2 and g x  1 for all x  0 we would again have x lim f x  Ý xÝ and lim g x  0 xÝ but, this time, we would have lim f x g x  Ý. xÝ Alternatively we could define f x  x and g x  1 x2 for all x  0 and obtain lim f x g x  0. xÝ These examples show that there is no way to define Ý 0 that will make the rule for products of limits work in the case in which one limit is zero and the other is Ý. 3. Thinking of the rule for quotients of limits lim xa fx gx  lim f x xa lxim g x a that you saw in elementary calculus, give some examples to show why the expression defined. Ý Ý should not be 4. Given that A and B are intervals and that A B , prove that the set A Þ B is an interval. Choose a number w A B. We define u  inf A Þ B and v  sup A Þ B . We see at once that u is the smaller of the two numbers inf A and inf B and that v is the larger of the two numbers sup A and sup B. Case 1: Assume that u  inf A and v  sup A: In this case it is clear that u  inf A inf B sup B sup A  v and A Þ B is one of the intervals u, v , u, v , u, v and u, v . Case 2: Assume that u  inf B and v  sup B. This case is similar to case 1. Case 3: Assume that u  inf A and v  sup B. In this case u  inf A inf B w sup A sup B  v Once again we can see that A Þ B must be one of the intervals u, v , u, v , u, v and u, v . To justify this assertion we need to see why every number x between u and v must belong to A Þ B. But if u  x  v then either u  x w or w  x  v. The inequality u  x w guarantees that x A and the inequality w  x  v guarantees that x B. Case 4: Assume that u  inf B and v  sup A. This case is similar to case 3. 5. Given that A, B and C are intervals and that the sets A B and B C are nonempty, prove that A Þ B Þ C is an interval. We deduce from Exercise 4 that A Þ B is an interval. Since AÞB C we deduce from Exercise 4 again that A Þ B Þ C is an interval. 92 Exercises on the Complex Number System 1. Find two complex numbers z  x  iy for which z 2  3  4i. Solution: The equation x  iy 2  3  4i gives us the system of two equations x2 which implies that x 4 3x 2 y2  3 2xy  4 4  0 from which we deduce that x  2. We see at once that x  iy  2  i . 2. Solve the quadratic equation z 2  2z  4  0. Why must the solutions of this equation be cube ro...
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This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.

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