67 3.1.9 Let f : D → R and 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 p √ lim f (x)= ℓ. x→x0 3.1.10 Prove that limx →0 xsin(1/x)= 0. 3.1.11 ▶ Let f : [0,1] → R be the function given by ( x, if x ∈ Q; f (x)= 1 − x, if x ∈ Qc . Determine which of the following limits exist. For those that exist fnd 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. Defnition 3.2.1 Let f ,g: D → R and let c be a constant. The functions f + g, fg, and cf are respectively defned as functions from D to R by ( f + g)(x)= f (x)+ g(x), ( fg)(x)= f (x)g(x), (cf )(x)= cf (x) e e for x ∈ D. Let D = {x ∈ D : g(x) ̸ = 0}. The function f /g is defned as a function from D to R by f f (x) (x)= , for x ∈ De. g g(x) Theorem 3.2.1 Let f ,g: D → R and let c ∈ R. Suppose x0 is a limit point of D and lim f (x)= ℓ, lim g(x)= m. x→x0 x→x0 Then (i) limx →x0 ( f + g)(x)= ℓ + m, (ii) limx →x0 ( fg)(x)= ℓm, (iii) limx →x0 (cf )(x)= cℓ, f ℓ (iv) lim (x)= provided that m ≠ 0. x→x0 g m
RkJQdWJsaXNoZXIy NTc4NTAz