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Unformatted text preview: Math1011  Learning Strategies Center 9/23/2009 Find the limits 2 1 lim x x   lim x x x 1 lim x x 1 lim x x lim cos x x lim 5 x x 5 3.5 h h Simplify: 1/6 h h Simplify: Math1011  Learning Strategies Center 9/23/2009 2 1 lim x x 2 1 lim x x =   lim x x x           lim lim 1 lim lim 1 lim , lim DNE x x x x x x x x x x x x x x x x x x x x x + + + = = = = As lim 1 lim x x 1 lim x x DNE (limit right limit left) 1 lim x x 1 lim x x = lim cos x x lim cos (oscillates between 1 and 1) x x D N E lim 5 x x 1 1 5 lim x = = 5 3.5 h h Simplify: 5 3.5 5 3.5 1.5 h h h h = = 1/6 h h Simplify: 1/6 1 (1/6) 5/6 h h h h = = (Doc #011r.07t) Math1011 Section 2.6 2 Given the graph of : At what points on the interval [5, 5] is discontinuous? Explain by indicating the type of discontinuity. At what points on its domain is discontinuous? Math1011 Section 2.6 2 Given the graph of : At what points on the interval [5, 5] is discontinuous? Explain by indicating the type of discontinuity. 2 (infinite discontinuity) 1 (removable discontinuity) 1 (jump discontinuity) 3 (removable discontinuity) At what points on its domain is discontinuous? 1 (jump discontinuity) 3 (removable discontinuity) Note: x = 2 and x = 1 are not included as these points are not in the domain of . (Doc #011.w.26.01t) Math1011 Section 2.6 3 Sketch the graph of the function and identify the type of discontinuity (x) has at x = 1. 2 4 3 1 ( ) 1 3 1 x x if x f x x if x + = = How could we redefine (x) to make it continuous? Math1011 Section 2.6 3 Sketch the graph of the function and identify the type of discontinuity (x) has at x = 1. 2 4 3 1 ( ) 1 3 1 x x if x f x x if x + = = Definition of Continuity A function is continuous at a number a if lim ( ) ( ) x a f x f a = 2 1 4 3 1 1 ( 1)( 3) 1 1 1 lim ( ) ? (1) lim ? (1) lim ? (1) lim 3 ? (1) 2 3 So, f(x) is not continuous at 1 Removable Discontinuity at x=1 x x x x x x x x x x f x f f f x f + How could we redefine (x) to make it continuous?...
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