Limits_and_continuity - LIMITS AND CONTINUITY 1 CHAPTER 2...

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LIMITS AND CONTINUITY 1 CHAPTER 2 LIMITS AND CONTINUITY 2.1 LIMITS One sided limits, Two sided limits, Limits of Polynomials and Rational Functions , Limits of Indeterminate form of 0 0 , Limits at Infinity, Horizontal Asymptotes, Limits of Rational Functions as x approaching infinity and limits of Trigonometric Functions. 2.2 CONTINUITY Definitions and conditions to be satisfied.
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LIMITS AND CONTINUITY 2 2.1 LIMITS One of the basic concepts of calculus is the limit. We need this concept to illustrate the continuity of a line and to form a first principal of derivatives. The continuous functions and the derivatives are very important to calculus. In this chapter we will discuss on an intuitive introduction of limits, one-sided limits, the relationship between one-sided and two-sided limits, limits of polynomials and rational functions, limits involving radicals, infinite limits, indeterminate form of type 0 0 and . Some simple limits on trigonometric functions, continuity of piecewise-defined functions, verticals and horizontal asymptotes are also will be focus on. If the value of ) ( x f can be made as close as we like to L by taking values of x sufficiently closed to a , then we write L x f a x = ) ( lim This expression is read “the limit of ) ( x f as x approaches a is L
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LIMITS AND CONTINUITY 3 Example 1 Find ( 29 1 lim 3 - x x Solution let f ( x ) = x – 1, from the graph, if x ĺ 3 , f ( x ) ĺ 2 Therefore ) ( lim 3 x f x = 2 Example 2 Find a) ( 29 1 2 lim 2 1 - + x x x d ) + - 2 lim 2 2 x x x b) + 2 2 lim 0 x x x e ) 4 0 1 lim - x x x c) - + - 1 1 lim 1 x x x f ( x ) f ( x ) = x – 1 2 x x =3
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LIMITS AND CONTINUITY 4 Solution a) ( 29 2 1 ) 1 ( 2 1 1 2 lim 2 2 1 = - + = - + x x x d) = = + - - = + - 0 4 2 2 ) 2 ( 2 lim 2 2 2 x x x The limit does not exist b) 0 2 0 2 0 ) 0 ( 2 2 2 lim 0 = = + = + x x x e ) = = - = - 0 1 0 1 0 1 lim 4 4 0 x x x . The limit does not exist. c) 0 2 0 1 1 1 1 1 1 lim 1 = - = - - + - = - + - x x x Example 3 Find a) x x 7 lim b ) -∞ 6 lim 2 x x Solution a) 0 7 7 lim = = x x b ) ( 29 = = - = -∞ 6 6 6 lim 2 2 x x The limits exist and equal to 0. The limit does not exist. ONE-SIDED LIMITS Some function as x approach a from the left side would not be the same as x approach a from the right side. If the values of ) ( x f can be made as close as we like to L by taking values of x sufficiently close to a (but greater than a ), then we write L x f a x = + ) ( lim This expression is read “the limit of ) ( x f as x approaches a from the right is L
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LIMITS AND CONTINUITY 5 If the values of ) ( x f can be made as close as we like to
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This note was uploaded on 12/01/2010 for the course SUKAN md taught by Professor Saipon during the Spring '10 term at Albany College of Pharmacy and Health Sciences.

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Limits_and_continuity - LIMITS AND CONTINUITY 1 CHAPTER 2...

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