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Unformatted text preview: ECON475 HW #1 Answers 1. , the sequence 1 Theorem 5.1 As . Furthermore, lim converges to a limit denoted by the symbol 1 . Proof Let / 1/ , 1 ; so . As lim 1 1 gets larger and goes to infinity, so does lim . Letting . (r is fixed.) Since . 1 1 , we find lim 1 1 2. In 1960; 5-year plan of Turkey indicated that per capita real income of Turkey will double in 20 years. i) What should be average growth rate of per capita real income for that? 1 2 20 ln 1 ln 2 3,52% 3. If , show that 3 We know that the derivative of f is (at point . ): Then, lim lim lim 3 3 lim 3 . 3 3 lim 4. Does Differentiability Continuity hold? No! i) (Differentiability Continuity) Proof Let : be differentiable at lim . Then, the limit exists and is the same for every sequence which converges to 0. If a function is differentiable at every point in its domain D, we say that the function is differentiable. 1 is continuous at if lim . Let lim . If f is differentiable, then at lim , lim 0 lim .0 0 exists. Let us now consider continuity. If lim that is continuous. lim Since lim , then we can say lim Therefore, ii) is differentiable, both limits exist. Thus, lim is continuous. (Continuity , 0, i.e. lim ) | |. Note that f is continuous everywhere, but it is not does not exist. 1 1 0 . 0 Consider : differentiable at 0 lim Why? 0 2 ...
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- Spring '10