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Unformatted text preview: 4. OneSided Limits and Special Trig Limits P. K. Lamm 09/08/11 (20:41) p. 1 / 8 Lecture Notes: 4. OneSided 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. LeftHand and RightHand 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 > , x 1 = 0 ( x 1) , x 1 < = ( x 1) , x ≥ 1 , 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 righthand limit and lefthand 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 lefthand limit of f ( x ) as x approaches x , written lim x → x f ( x ) = L, or f ( x ) → L as x → x , provided that the number f ( x ) may be made as close to L as desired by choosing x sufficiently close to x , x 6 = x , with x < x . 1 4. OneSided Limits and Special Trig Limits P. K. Lamm 09/08/11 (20:41) p. 2 / 8 Definition: The number L is the righthand limit of f ( x ) as x approaches x , written lim x → x + f ( x ) = L, or f ( x ) → L as x → x + , provided that the number f ( x ) may be made as close to L as desired by choosing x sufficiently close to x , x 6 = x , with x > x . For a lefthand limit, we’re interested in the height of the curve as x approaches x from the left, or from values of x smaller than x . For a righthand limit, we’re interested in the height of the curve as x approaches x from the right, or from values of x larger than x ....
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This note was uploaded on 04/02/2012 for the course MTH 132 taught by Professor Kihyunhyun during the Fall '10 term at Michigan State University.
 Fall '10
 KIHYUNHYUN
 Calculus, Limits

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