Quiz 2 - | (1 /x )-2 | < , or equivalently 1 /...

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Prof. Haiman Math 1A—Calculus Fall, 2004 Quiz 2 solution—version A Name Student ID Number 1. Calculate each limit, if it exists either as a number or as an infinite limit. If the limit doesn’t exist, say so. (a) lim x 2 x 3 - 4 x x - 2 = lim x 2 x ( x + 2)( x - 2) x - 2 = 8 . (b) lim x 1 1 x 2 - 1 doesn’t exist, since 1 / ( x 2 - 1) + as x 1 + , but 1 / ( x 2 - 1) → -∞ as x 1 - . 2. (a) The fact that lim x 1 / 2 1 x = 2 means that for every ± > 0, there exists a δ > 0 such that some condition holds. State that condition (as it applies to this specific limit). 0 < | x - 1 / 2 | < δ implies
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Unformatted text preview: | (1 /x )-2 | &lt; , or equivalently 1 / 2- &lt; x &lt; 1 / 2 + , x 6 = 1 / 2 implies 2- &lt; 1 /x &lt; 2 + . (b) Find a that veries the required condition if = 0 . 1 . To get 1 . 9 &lt; 1 /x &lt; 2 . 1, need 1 / (2 . 1) &lt; x &lt; 1 / (1 . 9), so can be any positive number less than or equal to the smaller of 1 / 2-1 / (2 . 1) = 1 / 42 and 1 / (1 . 9)-1 / 2 = 1 / 38, that is, any &lt; 1 / 42. For example, = . 02 would do. 1...
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This note was uploaded on 03/08/2010 for the course MATH 1A taught by Professor Wilkening during the Fall '08 term at University of California, Berkeley.

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