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Unformatted text preview: Math021, week 4 One-Sided Limits Definition 4.1 Let f be a function and c be a number. If f ( x ) approaches a number L when x is bigger (smaller) than but close to the number c , we say that f ( x ) approaches (tends to) L as x approaches c from the right (left). In symbol lim x → c + f ( x ) = L ( lim x → c- f ( x ) = L ) . Theorem 4.2 Let f be a function and c be a number. lim x → c f ( x ) exists if and only if both lim x → c + f ( x ) and lim x → c- f ( x ) exist and lim x → c + f ( x ) = lim x → c- f ( x ) . In this case all the three limits men- tioned are the same. proof: Not as difficult as the previous ones but we will still omit it. Example 4.3 Let f ( x ) = ‰ x when x ≥ 1- x 2 when x < Determine if the limit lim x → f ( x ) exists. solution: Since f ( x ) = x whenever x > 0, lim x → + f ( x ) = lim x → + x = 0 . On the other hand f ( x ) = 1- x 2 whenever x < 0, lim x →- f ( x ) = lim x →- 1- x 2 = 1 . We see that lim x → + f ( x ) is different from lim x →- f ( x ). The limit lim x → f ( x ) does not exist by the previous theorem. 1 Example 4.4 Determine if the limit lim x → | x | exists. solution: Since | x | = x whenever x > 0, lim x → + | x | = lim x → + x = 0. On the other hand | x | =- x whenever x < 0, lim x →- | x | = lim x →-- x = 0. We see that both lim x → + | x | and lim x →- | x | exist and they are the same. The limit lim x → | x | exists and is 0. Remark 4.5 The results in theorem 3.14 and the sandwich theorem hold for one-sided limits. Theorem 4.6 lim x → sin x x = 1 proof: First of all, suppose that x > 0. Let O be the center of the unit circle. X is the point on the unit circle whose inclination is x . A is the intersection of the positive x-axis and the unit circle. The line OX extends to meet the tangent of the unit circle at A in the point Y . The foot of perpendicular from X and Y to the x-axis are X , Y respectively. Then, area of Δ OXX ≤ area of the sector XOA ≤ area of Δ OY Y ....
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This note was uploaded on 02/13/2012 for the course MATH 021 taught by Professor Luxnu during the Fall '10 term at HKUST.
- Fall '10