2 (4). By (3), it is easy to see that 360° 0<S n+1.3 In ont :2n+1.3.sin -0600 360 cos on+l 360° 3 (5). To prove that (Sn) is a Cauchy sequence 0<S+2-Sn<∑|Sk+1-58|<∑ (6). The conclusion is obvious from (5) 补充习题 、证明:任何点集的内点全体是开集 2、设limf(x)=a,且f(x)在xo有定义。问在x→xo的过程中,x可否取到xo?是否必有 a=f(zo)? 试用ε-δ语言来叙述当工→xo时f(x)不收敛于a 4、用分析定义证明 31 1 (4)lim (x-2)(x-1) (5 (7)lim x→∞x+1=∞ 5、下列命题是否正确?若正确,请给岀证明;若不正确,请举出反例。 (1)limf(x)=a充分必要条件是imf(x)=|l (2)若mnfx)=a则inUf()2=a2 (3)若limf()=a,则limf(x)=a (4)若Hn。f(x)与imn(()+9(a)都存在,则1in,9()存在 (5)若imf(x)与lim(f(x)g(x)都存在,则limg(x)存在 (6)若在xo的某邻域内f(x)>0,并且limf(x)=a,那么必有a>02 (4). By (3), it is easy to see that 0 < Sn+1 − Sn =2n+1 · 3 · sin 360◦ 2n+1 · 3 − 2n · 3 · sin 360◦ 2n · 3 =2n+1 · 3 · sin 360◦ 2n+1 · 3  1 − cos 360◦ 2n+1 · 3  =2n+1 · 3 · sin 360◦ 2n+1 · 3 · 2 · sin2  360◦ 2n+2 · 3  < 9 4 · 2 3 √3 n+1 (5). To prove that {Sn} is a Cauchy sequence: 0 < Sn+p − Sn < n  +p k=n |Sk+1 − Sk| < 9 4 · n  +p k=n 2 3 √ 3 k+1. (6). The conclusion is obvious from (5).  1  !  !"!# 2 " limx→x0 f(x) = a, Æ f(x) # x0 $$%#&# x → x0 %&' x '(() x0 Æ!(*$ a = f(x0)? 3 )* ε − δ +,+-., x → x0 / f(x) -0.1 a. 4 *$% (1) limx→1 x − 3 x2 − 9 = 1 2 ; (2) limx→3 x − 3 x2 − 9 = 1 6 ; (3) limx→1 x − 1 √x − 1 = 2; (4) limx→1 (x − 2)(x − 1) x − 3 = 0; (5) limt→1 t(t − 1) t2 − 1 = 1 2 ; (6) limx→∞ x − 1 x + 2 = 1; (7) limx→3 x x2 − 9 = ∞; (8) limx→∞ x2 + x x + 1 = ∞. 5 2/3!(45Æ64570126-4573145# (1) limx→x0 f(x) = a *896! limx→x0 |f(x)| = |a|; (2) 6 limx→x0 f(x) = a, : limx→x0 [f(x)]2 = a2; (3) 6 limn→∞ f( 1 n) = a, : lim x→0+ f(x) = a; (4) 6 limx→x0 f(x) ; limx→x0 (f(x) + g(x)) 78#: limx→x0 g(x) 8#2 (5) 6 limx→x0 f(x) ; limx→x0 (f(x)g(x)) 78#: limx→x0 g(x) 8#2 (6) 6# x0 <9= f(x) > 0 :Æ lim x→x− 0 f(x) = a >?*$ a > 0.