1007slides_Part2

1007slides_Part2 - Math 1007, Fall 2006 (These slides...

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Math 1007, Fall 2006 (These slides replace neither the text book nor the lectures.) Part II: Limits 11. Limits (17-19) 12. Properties of Limits (20-21) (22-26) 14. One Sided Limits (27-28) 15. Continuity (29-31) 16
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11. Limits (2.2) Consider the function f ( x ) = x 2 . We wish to determine the value that the function f approaches when x is close to 2. Approaching from the left: x < 2 x 1.75 1.9 1.99 1.999 f ( x ) 3.0625 3.61 3.9601 3.996001 Approaching from the right: x > 2 x 2.25 2.1 2.01 2.001 f ( x ) 5.0625 4.41 4.0401 4.004001 Since f approaches 4 as x approaches 2, we write lim x 2 f ( x ) = lim x 2 x 2 = 4 . 17
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De¯nition: The function f has the limit L as x ap- proaches a , written lim x a f ( x ) = L if the value of f ( x ) can be made as close to the number L as we please by taking x su±ciently close to (but not equal to) a . Example 1: Consider g ( t ) = 4( t 2 4) t 2 . Note that g is not de¯ned at t = 2. t 1.9 1.99 1.999 2.001 2.01 2.1 g ( t ) 15.6 15.96 15.996 16.004 16.04 16.4 Since g ( t ) approaches 16 if t approaches 2, lim t 2 g ( t ) = lim x 2 4( t 2 4) t 2 = 16 . Hence the limit at a may exist even if the function is not de¯ned at a . 18
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Example 2: Consider g ( x ) = b x + 2 x n = 1 1 x = 1 From the graph, we see that with x near 1 (but not equal to 1), g ( x ) is near 3 so lim x 1 g ( x ) = 3 . Note that g (1) = 1, so the value of the function has no bearing on the value of the limit. Example 3:
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This note was uploaded on 03/22/2010 for the course MATH 1007 taught by Professor Unknown during the Fall '07 term at Carleton.

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1007slides_Part2 - Math 1007, Fall 2006 (These slides...

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