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lec04 - 4 One-Sided Limits and Special Trig Limits Lecture...

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4. One-Sided Limits and Special Trig Limits P. K. Lamm 09/08/11 (20:41) p. 1 / 8 Lecture Notes: 4. One-Sided Limits and Special Trig Limits These classnotes are intended to be supplementary to the textbook and are necessarily limited by the time allotted for classes. For full and precise statements of definitions and theorems, as well as material covering other topics and examples, please consult the textbook. 1. Left-Hand and Right-Hand Limits Example 1.1(a): Evaluate: lim x 1 | x - 1 | x - 1 . Recalling the definition of the absolute value, we have | x - 1 | = ( x - 1) , x - 1 > 0 0 , x - 1 = 0 - ( x - 1) , x - 1 < 0 = ( x - 1) , x 1 0 , x = 1 - ( x - 1) , x < 1 . Thus, | x - 1 | x - 1 = ( x - 1) x - 1 , x > 1 undefined , x = 1 - ( x - 1) ( x - 1) , x < 1 . = 1 , x > 1 undefined , x = 1 - 1 , x < 1 . Because there is no one number that y = | x - 1 | / ( x - 1) approaches as x 1 from either side of 1, we have that lim x 1 | x - 1 | x - 1 is undefined . But this doesn’t tell the entire story. We can use the idea of right-hand limit and left-hand limit to give more information about what is happening to this curve as x approaches 1 from either side of 1: Definition: The number L is the left-hand limit of f ( x ) as x approaches x 0 , written lim x x - 0 f ( x ) = L, or f ( x ) L as x x - 0 , provided that the number f ( x ) may be made as close to L as desired by choosing x sufficiently close to x 0 , x 6 = x 0 , with x < x 0 . 1
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4. One-Sided Limits and Special Trig Limits P. K. Lamm 09/08/11 (20:41) p. 2 / 8 Definition: The number L is the right-hand limit of f ( x ) as x approaches x 0 , written lim x x + 0 f ( x ) = L, or f ( x ) L as x x + 0 , provided that the number f ( x ) may be made as close to L as desired by choosing x sufficiently close to x 0 , x 6 = x 0 , with x > x 0 . For a left-hand limit, we’re interested in the height of the curve as x approaches x 0 from the left, or from values of x smaller than x 0 . For a right-hand limit, we’re interested in the height of the curve as x approaches x 0 from the right, or from values of x larger than x 0 . Example 1.1(b): We return to the last example, only now looking at left-hand and right-hand limits. From the work in the last example, we have lim x 1 - | x - 1 | x - 1 = - 1 , lim x 1 + | x - 1 | x - 1 = 1 .
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