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PART III. FUNCTIONS: LIMITS AND CONTINUITY

PART III. functions : LIMITS AND LIMITS OF FUNCTIONSThis chapter is concerned with functionsf:D RwhereDis a nonempty subset ofR. That is, we will be considering real-valued functions of a real variable. The setDiscalled :D Rand letcbe an accumulation point ofD. A numberLis thelimit offatcif to each >0there exists a >0such that|f(x) L|< wheneverx Dand0<|x c|< .This definition can be stated equivalently as :D Rand letcbe an accumulation point ofD. A numberListhelimit offatcif to each neighborhoodVofLthere exists a deleted neighborhoodUofcsuch thatf(U D) cf(x)= :(a) limx 2(x2 2x+ 4) = 12.(b) limx 2x2 4x 2=4.(c) limx 3x2+3x+5x 3does not exist.(d) limx 1|x 1|x 1does not :Letf(x)=4x 5. Prove that limx 3f(x)= :Let >0.|f(x) 7|=|(4x 5) 7|=|4x 12|=4|x 3|.Choose = /4. Then|f(x) 7|=4|x 3|<4 4= whenever 0<|x 3|<.

PART III. FUNCTIONS: LIMITS AND CONTINUITY III.1. LIMITS OF FUNCTIONS This chapter is concerned with functions f: D → R where D is a nonempty subset of R. That is, we will be considering real-valued functions of a real variable. The set D is called the domain of f. Definition 1. Let f: D → R and let c be an accumulation point of D. A number L

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