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SECTION 2.4 Continuity & One-Sided Limits Discontinuous v. Continuous Formal Definition of Continuity Definition of Continuity (p. 90) Continuity at a Point: A function π is continuous at π if the following three conditions are met. 1. π(π) is defined. 2. lim π(π₯) exisits. π₯βπ 3. lim π(π₯) = π(π). π₯βπ 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. A function that is continuous on the entire real line (-β,β) is everywhere continuous. Two Types of Discontinuity β’ Removable & Non-Removable β’ A discontinuity at π is called removable if π can be made continuous by appropriately defining (or redefining) π π . One-Sided Limits 1. 2. lim+ π(π₯) π₯βπ limβ π(π₯) π₯βπ βthe limit of π as π₯ approaches π from the rightβ βthe limit of π as π₯ approaches π from the leftβ When Does a Limit Exist? Theorem 2.10 The Existence of a Limit (p. 93) Let π be a function and let π and πΏ be real numbers. The limit of π(π₯) as π₯ approaches π is πΏ if and only if lim+ π(π₯) = πΏ π₯βπ and lim π(π₯) = πΏ. π₯βπ β Example 1 (#6) a. b. lim+ π(π₯) = π₯βπ limβ π(π₯) = π₯βπ c. lim π(π₯) = π₯βπ Calculating One-Sided Limits 1. Plug in the x-value that you are approaching. 2. If you get a real number, then thatβs your limit. 3. If not, try some algebra to see if things can cancel. 4. If that doesnβt work, then plug in x-values extremely close to the number you are approaching or graph it. Example 2 Find the limit if it exists. If not, explain. a. 2βπ₯ lim+ 2 π₯β2 π₯ β4 b. π₯β10 lim + π₯β10 π₯β10 c. lim + βπ₯β0 π₯+βπ₯ 2 +π₯+βπ₯β π₯ 2 +π₯ βπ₯ Example 2 (cont.) Find the limit if it exists. If not, explain. a. b. π₯, lim+ π(π₯) where π π₯ = 1 β π₯, π₯β1 π₯β€1 π₯>1 π₯ 2 β 4π₯ + 6, π₯<2 lim π(π₯) where π π₯ = π₯β2 βπ₯ 2 + 4π₯ β 2, π₯ β₯ 2 Continuity on a Closed Interval Definition of Continuity on a Closed Interval (p. 93) A function π is continuous on the closed interval [a,b] if it is continuous on the open interval (a,b) and lim+ π(π₯) = π(π) π₯βπ and limβ π(π₯) = π(π). π₯βπ The function π is continuous from the right at π and continuous from the left at π. Pictorial Representation Example 3 Discuss the continuity of the function on the closed interval. a. π π‘ = 2 β 9 β π‘2 b. π π₯ = 1 π₯ 2 β4 [β2, 2] [β1, 2] Example 4 Find the π₯-values where π is discontinuous. If any, which are removable? 3 π₯β2 a. π π₯ = b. π π₯ = π₯ 2 β 2π₯ + 1 c. d. π π₯ = π₯β3 π₯ 2 β9 π π₯ = π₯β3 π₯β3