CalcNotes0206-page7

CalcNotes0206-page7 - (possibly excluding 0) so that we can...

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(Section 2.6: The Squeeze (Sandwich) Theorem) 2.6.7 Example 3 (Limits are Local) The open x -interval ± 1, 1 () contains 0. On this interval, As x ± 0, x 6 ± 0 ± ² x 4 Therefore, ± 0 ± ² x 2 ± 0 ± ³ 1 < x < 1 () Note : Observe that 1 2 ± ² ³ ´ µ 4 = 1 16 , 1 2 ± ² ³ ´ µ 2 = 1 4 , and 1 16 < 1 4 . We conclude: lim x ± 0 x 4 = 0 , which we knew already. We do not need the compound inequality to hold true for all nonzero values of x . We only need it to hold true on some open x -interval containing 0
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Unformatted text preview: (possibly excluding 0) so that we can immediately discuss the two-sided limit lim x x 4 . This is because Limits are Local. As seen below, the graphs of y = x 6 and y = x 2 squeeze (from below and above, respectively) the graph of y = x 4 on the x-interval 1, 1 ( ) . In fact, this is the case on the closed x-interval 1, 1 . (Figure 2.6.b)...
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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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