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Calculus Ch1 Review – Limits Behavior Associated with Nonexistence of a Limit 1. f(x) approaches a different number from the right side of c than it approaches from the left side. 2. f(x) increases or decreases without bound as x approaches c. 3. f(x) oscillates between two fixed values as x approaches c. Theorem 1.1: Some Basic Limits. Let b and c be real numbers and let n be a positive integer. 2.lim x c 1.lim b b 3.lim xn cn x c x c xc Theorem 1.2 Properties of Limits. Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits. lim f ( x) L and lim g ( x) K x c x c 1. Scalar multiple: lim bf ( x) bL x c 2. Sum or Difference: lim f ( x) g ( x) L K xc 3. Product: 4. Quotient 5. Power: lim f ( x) g ( x) LK xc f ( x) L provided K≠0 lim x c g ( x ) K n lim f ( x) Ln x c Strategies for Finding Limits as x approaches c 1. Try direct substitution if the function is a polynomial, rational, or trig. 2. If direct substitution causes division by zero, then a. try factoring and if a common factor cancels the denominator then direct substitution b. try rationalizing the numerator 3. If these both fail, then look at a graph and a table of values as x approaches c Theorem 1.9 Two Special Trig Limits sin x 1 cos x 1.lim 1 2.lim 0 x 0 x 0 x x Definition of Continuity at a Point:A function f is continuous at c if the following 3 conditions are met. 1. f(c) is defined. 2. lim f ( x ) exists 3. lim f ( x ) = f(c) x c x c Continuity on an open interval: A function is continuous on an open interval (a,b) if it is continuous at each point in the interval. If a function f is defined on an interval I (except possibly at c), and f is not continuous at c, then f is said to have a discontinuity at c. A discontinuity is said to be removable if f can be made continuous by appropriately defining (or redefining) f(c). Otherwise, the discontinuity is said to be nonremovable. Theorem 1.10 The Existence of a Limit: Let f be a function and let c and L be real numbers. The limit of f(x) as x approaches c is L if and only if lim f ( x) L and lim f ( x) L x c x c A function f is continuous on the closed interval [a,b] if it is continuous on the open interval (a,b) and lim f ( x) f (a) and lim f ( x) f (b) x b x a Intermediate Value Theorem: If f is continuous on the closed interval [a,b], f(a) ≠ f(b), and k is any number between f(a) and f(b), then there is at least one number c in [a,b] such that f(c) = k. 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. bf 2. f±g 3. fg 4. f/g Definition of a vertical asymptote: If f(x) approaches infinity (or negative infinity) as x approaches c from the right or the lefet, then the line x = c is a vertical asymptote of the graph of f. Theorem 1.14 about Vertical Asymptotes: Let f and g be continuous on an open interval containing c. If f(c) ≠ 0 and g(c) = 0, and there exists an open interval containing c such that g(x) ≠ 0 for all x ≠ c in f ( x) the interval, then the graph of the function given by h( x) has a vertical asymptote at x = c. g ( x) Theorem 1.15 Properties of Infinite Limits: Let c and L be real numbers and let f and g be functions such that lim f ( x) and lim g ( x) L x c 1. lim f ( x) g ( x) x c xc 2. lim f ( x) g ( x) if L > 0 x c 3. lim f ( x) g ( x) if L < 0 xc g ( x) 4. lim 0 x c f ( x) p. 91 9,10,15,17,27-31, 35,37,38,53