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ISM_T11_C02_A - CHAPTER 2 LIMITS AND CONTINUITY 2.1 RATE OF...

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CHAPTER 2 LIMITS AND CONTINUITY 2.1 RATE OF CHANGE AND LIMITS 1. (a) Does not exist. As x approaches 1 from the right, g(x) approaches 0. As x approaches 1 from the left, g(x) approaches 1. There is no single number L that all the values g(x) get arbitrarily close to as x 1. Ä (b) 1 (c) 0 2. (a) 0 (b) 1 (c) Does not exist. As t approaches 0 from the left, f(t) approaches 1. As t approaches 0 from the right, f(t) approaches 1. There is no single number L that f(t) gets arbitrarily close to as t 0. Ä 3. (a) True (b) True (c) False (d) False (e) False (f) True 4. (a) False (b) False (c) True (d) True (e) True 5. lim does not exist because 1 if x 0 and 1 if x 0. As x approaches 0 from the left, x 0 Ä x x x x x x x x x x k k k k k k œ œ œ œ approaches 1. As x approaches 0 from the right, approaches 1. There is no single number L that all x x x x k k k k the function values get arbitrarily close to as x 0. Ä 6. As x approaches 1 from the left, the values of become increasingly large and negative. As x approaches 1 " x 1 from the right, the values become increasingly large and positive. There is no one number L that all the function values get arbitrarily close to as x 1, so lim does not exist. Ä x 1 Ä " x 1 7. Nothing can be said about f(x) because the existence of a limit as x x does not depend on how the function Ä ! is defined at x . In order for a limit to exist, f(x) must be arbitrarily close to a single real number L when ! x is close enough to x . That is, the existence of a limit depends on the values of f(x) for x x , not on the ! ! near definition of f(x) at x itself. ! 8. Nothing can be said. In order for lim f(x) to exist, f(x) must close to a single value for x near 0 regardless of x 0 Ä the value f(0) itself. 9. No, the definition does not require that f be defined at x 1 in order for a limiting value to exist there. If f(1) œ is defined, it can be any real number, so we can conclude nothing about f(1) from lim f(x) 5. x 1 Ä œ 10. No, because the existence of a limit depends on the values of f(x) when x is near 1, not on f(1) itself. If lim f(x) exists, its value may be some number other than f(1) 5. We can conclude nothing about lim f(x), x 1 x 1 Ä Ä œ whether it exists or what its value is if it does exist, from knowing the value of f(1) alone.
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68 Chapter 2 Limits and Continuity 11. (a) f(x) x /(x 3) œ * a b # x 3.1 3.01 3.001 3.0001 3.00001 3.000001 f(x) 6.1 6.01 6.001 6.0001 6.00001 6.000001 x 2.9 2.99 2.999 2.9999 2.99999 2.999999 f(x) 5.9 5.99 5.999 5.9999 5.99999 5.999999 The estimate is lim f(x) 6. x Ä $ œ (b) (c) f(x) x 3 if x 3, and lim (x 3) 3 3 6. œ œ œ Á œ œ x 9 x 3 x 3 (x 3)(x 3) x Ä $ 12. (a) g(x) x / x 2 œ # a b Š È # x 1.4 1.41 1.414 1.4142 1.41421 1.414213 g(x) 2.81421 2.82421 2.82821 2.828413 2.828423 2.828426 (b) (c) g(x) x 2 if x 2, and lim x 2 2 2 2 2. œ œ œ Á œ œ x 2 x 2 x 2 x 2 x 2 È Š ‹ Š È È Š È È È È È È È Š x Ä # È 13. (a) G(x) (x 6)/ x 4x 12 œ a b # x 5.9 5.99 5.999 5.9999 5.99999 5.999999 G(x) .126582 .1251564 .1250156 .1250015 .1250001 .1250000 x 6.1 6.01 6.001 6.0001 6.00001 6.000001 G(x) .123456 .124843 .124984 .124998 .124999 .124999
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Section 2.1 Rates of Change and Limits 69 (b) (c) G(x) if x 6, and lim 0.125. œ œ œ Á œ œ œ x 6 x 6 x 4x 12 (x 6)(x 2) x x 2 2 8 " " " " # ' a b x Ä ' 14. (a) h(x) x 2x 3 / x 4x 3 œ a b a b # # x 2.9 2.99 2.999 2.9999 2.99999 2.999999 h(x) 2.052631 2.005025 2.000500 2.000050 2.000005 2.0000005 x 3.1 3.01 3.001 3.0001 3.00001 3.000001 h(x) 1.952380 1.995024 1.999500 1.999950 1.999995 1.999999 (b) (c) h(x) if x 3, and lim 2. œ œ œ Á œ œ œ x 2x 3 x 1 x 1 3 1 4 x 4x 3 (x 3)(x 1) x
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