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# Section 2.4 - • A continuity is nonremovable if the limit...

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Chapter Two: Limits Section 2.4 One – Sided Limits: ( ) lim x c f x - is the limit as x approaches c from the left . ( ) lim x c f x + is the limit as x approaches c from the right. ( ) ( ) ( ) x 0 x 0 x 0 limf x 0, limf x 2,limf x DNE - + = = = Existence of the Limit: ( ) ( ) ( ) lim lim lim x c x c x c f x L f x L f x - + = = = , for c and L real numbers and f a function. Definition: A function ( ) f x is continuous at c if the following three conditions are met: 1. ( ) f c is defined. 2. ( ) lim x c f x exists. 3. ( ) ( ) lim x c f x f c =

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A function is continuous on an open interval (a, b) if it is continuous at each point in the interval. A function that is continuous on the entire real line ( ) , -∞ ∞ is continuous everywhere. Definition: A point x = c where a function is not continuous is called a discontinuity . There are two types of discontinuities: removable and nonremovable. A continuity is removable if the limit exists. A continuity is nonremovable if the limit does not exist.
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Unformatted text preview: • A continuity is nonremovable if the limit does not exist. Properties of Continuity • If b is a real number and f and g are continuous at x = c, then the following functions are also continuous at c: 1. Scalar Multiple: bf 2. Sum and Difference: f g ± 3. Product: fg 4. Quotient: ( ) , f g c g ≠ • The following functions are continuous at every point in their domains: 1. Polynomials 2. Rational 3. Radical 4. Trigonometric, Exponential and Logarithmic • If g is continuous at c and f is continuous at ( ) g c , then the composite function given by ( )( ) ( ) ( ) f g x f g x = r is continuous at c. Intermediate Value Theorem • If f is continuous on the closed interval [a, b] and k is any nonzero number between ( ) f a and ( ) f b , then there is at least one number c in [a, b] such that ( ) f c k = ....
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