CalcNotes0203-page30

CalcNotes0203-page30 - • Then x k ± ² as x ± ² lim x...

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(Section 2.3: Limits and Infinity I) 2.3.30 The following variation of the contrapositive of the preceding if-then statement is also true: If Dx () does not approach a real nonzero constant, then 1 Dx () does not approach a real nonzero constant, either. lim x ±² 1 sin x = lim x ±² csc x does not exist (DNE), so the 1 DNE limit “situation” can yield a nonexistent limit (“DNE”). • It is possible to obtain a limit that is 0, even if Dx () is not approaching ± , nor ±² . For example, let: Dx () = x ,i f x is a rational value ± x ,i f x is an irrational value ² ³ ´ . Then, lim x ±² Dx () does not exist (DNE). However, lim x ±² 1 Dx () = 0 . If we extend the real number system to the real projective line, in which we collapse together and identify ± and ±² , then we are effectively dealing with the generic Limit Form 1 ± , which yields 0 as a limit. 4. Justification for “Long-Run” Limit Rules for c x k . Assume that c is a real constant (i.e., any arbitrary real constant), and k is a positive rational constant.
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Unformatted text preview: • Then, x k ± ² as x ± ² . lim x ±² c x k has Limit Form c ± , which implies that the limit is 0 (regardless of whether c is positive, negative, or 0). • Also, x k ± ²³ as x ± ²³ , if x k is defined as a real quantity whenever x < 0 . Then, lim x ±²³ c x k has Limit Form c ±² , which implies that the limit is 0. If x k is undefined as a real quantity whenever x < 0 , then lim x ±²³ x k does not exist (DNE), and lim x ±²³ c x k does not exist (DNE), even if c = 0 . 5. Irrational Exponents; Roots of Negative Real Numbers. It is true that x k ± ² and lim x ±² c x k = 0 (for any real constant c ) for any positive real constant k . But what if k is irrational? For example, if k = ± , then how do we define something like 2 when x = 2? Remember that = 3.14159. ... Consider the corresponding sequence: 2 3 = 8 2 3.1 = 2 31 10 = 2 31 10 ± 8.57419 2 3.14 = 2 314 100 = 2 157 50 = 2 157 50 ± 8.81524 ±...
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This note was uploaded on 12/29/2011 for the course MATH 150 taught by Professor Sturst during the Fall '10 term at SUNY Stony Brook.

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