Introduction to Mathematical Analysis I - 3rd Edition

67 3.1.9 Let f : D→Rand let x0 be a limit point of D. Suppose f (x) ≥0 for all x ∈D. Prove that if limx→x0 f (x) =ℓ, then lim x→x0 p f (x) =√ℓ. 3.1.10 Prove that limx→0xsin(1/x) =0. 3.1.11 ▶Let f : [0,1] →Rbe the function given by f (x) =( x, if x ∈Q; 1−x, if x ∈Qc. Determine which of the following limits exist. For those that exist find their values. (a) limx→1/2 f (x). (b) limx→0 f (x). (c) limx→1 f (x). 3.2 Limit Theorems Here we state and prove various theorems that facilitate the computation of general limits. Definition 3.2.1 Let f ,g: D→Rand let c be a constant. The functions f +g, f g, and c f are respectively defined as functions fromDtoRby (f +g)(x) =f (x)+g(x), (f g)(x) =f (x)g(x), (c f )(x) =c f (x) for x ∈D. Let e D={x ∈D: g(x)̸ =0}. The function f /gis defined as a function frome DtoRby f g (x) = f (x) g(x) , for x ∈ e D. Theorem 3.2.1 Let f,g: D→Rand let c ∈R. Suppose x0 is a limit point of Dand lim x→x0 f (x) =ℓ, lim x→x0 g(x) =m. Then (i) limx→x0(f +g)(x) =ℓ+m, (ii) limx→x0(f g)(x) =ℓm, (iii) limx→x0(c f )(x) =cℓ, (iv) lim x→x0 f g (x) = ℓ m provided that m̸=0.

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