L08 - Trigonometric Inverse Trigonometric Exponential...

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L8 – Continuity Def – A function f is continuous at a number a if lim x a f ( x ) = f ( a ). If not, then f is discontinuous at a . 1. 2. 3. Graphically – a continuous function has no jumps, holes, or gaps. 1
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There are two types of discontinuities. Removable Nonremovable Jump Infinite 2
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Discuss the continuity of f ( x ) = x 2 - 1 x 3 - x . What types of discontinuities does f ( x ) have? Could we define f ( x ) to make it continuous? 3
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4
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Discuss the continuity of f ( x ) = x 2 - 1 x < 2 3 x = 2 5 2 x - 1 x > 2 5
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Def – A function f is continuous from the right at a if Continuous from the left at a if A function is continuous on an interval if 6
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Find the value of c which will make f ( x ) continuous. f ( x ) = 2 x x ≥ - 1 3 x + c x < - 1 7
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Discuss the continuity of f ( x ). Determine any removable or nonremovable discon- tinuities. f ( x ) = x - 1 x < 0 x 2 + 1 0 x 2 7 - x x > 2 8
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Thm – If f and g are continuous at a and c is a constant, then Thm – Polynomials Rational Root
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Unformatted text preview: Trigonometric Inverse Trigonometric Exponential Logarithmic 9 Where is g ( x ) = x-1 + x-3 x 2 continuous? If f is continuous at b and lim x c g ( x ) = b , then lim x c f ( g ( x )) = If g is continuous at a and f is continuous at g ( a ), then ( f g )( x ) is continuous at a . i.e. 10 Discuss the limits based on continuity of the functions. lim x e x 2 +3 x-1 lim x 2 cos( x-cos( x )) 11 An important property of continuous functions is the Intermediate Value Theorem. Suppose f is continuous on [ a,b ] and N is any number between f ( a ) and f ( b ), then 12 Use the IMVT to prove that f ( x ) = x 3 + 3 x-2 has a zero in [-1 , 1]. 13 Discuss the continuity of f ( x ) = 5-x x &gt; 1 x 2 + 1 x 1 14...
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L08 - Trigonometric Inverse Trigonometric Exponential...

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