1873_solutions

S a that satisfies the inequality x a 3 holds

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Given x 3, the inequality f x  w says that 1  w. |x 3 | We can’t simply turn these expressions over because w need not be positive but we can make the observation that the inequality 1 w |x 3 | will certainly hold when 1  |w |  1 |x 3 | which says that 1. |x 3 |  |w |  1 We therefore define   |w11 and observe that the condition f x  w will hold whenever x 3 and | |x 3 |  . fx  3. Given that 1 x3 for all numbers x 3, explain why f has an infinite limit from the left at 3 and also has an infinite limit from the right at 3 but does not have a limit at 3. The reason f has no two-sided limit at 3 is that the limits of f at 3 from the left and from the right are not equal to each other. In fact, the limit from the right is Ý and the limit from the left is Ý. To see why the f x Ý as x 3 , suppose that w is any real number. Given x  3, the inequality f x  w says that 1  w. x3 We can’t simply turn these expressions over because w need not be positive but we can make the observation that the inequality 1 w x3 fx  189 will certainly hold when 1 x  |w |  1 3 which says that 1. |w |  1 and observe that the condition f x  w will hold whenever 3 x We therefore define   |w11 | 3  x  3. To see why the f x Ý as x f x  w says that 3 , suppose that w is any real number. Given x  3, the inequality 1 x 3 w which we can express as 1 w 3x We can’t simply turn these expressions over because w need not be positive but we can make the observation that the inequality 1 w 3x will certainly hold when 1  |w |  1 x3 which says that 1. 3 x |w |  1 We therefore define   |w11 and observe that the condition f x  w will hold whenever | 3   x  3. 4. Prove that x3 8 x2  x 6 as x Ý. We begin by observing that x 2  x Ý 6  0 whenever x  2. Given any number x  2 we have 8  x 2  2x  4  x 2  x. x 2x x3 6 x Now suppose that w is any real number and define v to be the larger of the two numbers 2 and w. The inequality x3 8  w 2x 6 x will hold whenver x  v. x3 5. Prove that x 4 4x 3 x 2  x  7 x 3 2x 2 2x 3 Ý as x Ý. Given any number x we have x3 2x 2 2x 3 x 3 x2  x  1 2x 2 2x 3 will be positive whenever x  3. Now given any number x  3 we have x 4 4x 3 x 2  x  7  x 4 4x 3 x 2  x 4 4x 3 x 3  x 5  x 5 . x 3 x3 x3 x 3 2x 2 2x 3 Now suppose that w is any real number. We define v to be the larger of the two numbers 3 and w 5 and observe that the inequality 3 and so x 3 190 x 4 4x 3 x 2  x  7  w x 3 2x 2 2x 3 will hold whenever x  v. 6. Prove that 3x 2  x 1 5x 2  4 as x Ý. Given any number x we have 3x 2  x 1 5x 2  4 Whenever x  17 5 |5x 17| . 5 5x 2  4  we observe that 3x 2  x 1 5x 2  4 Now suppose that 3 5 3 5 |5x 5 5x 2  5x 2 5 5x 3 5  0. As long as x  17 5  17| 4 17 4  5x  1 . 5x 25x 2 , the inequality 3 x 1 5 5x 2  4 1 1 will hold whenever 5x  which says that x  5 . We define v to be the larger of the two numbers 17 1 and 5 and oberve that the inequality 5 3x 2 holds whenever x  v. 3x 2...
View Full Document

This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.

Ask a homework question - tutors are online