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 →0 xsin(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, fg, and cf are respectively defined as functions fromDto Rby (f +g)(x)=f (x)+g(x), (fg)(x)=f (x)g(x), (cf )(x)=cf (x) for x ∈D. Let eD={x ∈D: g(x)̸=0}. The function f /g is defined as a function fromeDtoRby f g (x)= f (x) g(x) , for x ∈ eD. 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(fg)(x)=ℓm, (iii) limx →x0(cf )(x)=cℓ, (iv) lim x→x0 f g (x)= ℓ m provided that m̸=0.
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