3 6、证明: (1)f(x)=a台→,imf()=,im、f(x)=a (2)imf(x)=a←÷limf(x)=limf(x)=a 7、证明:函数极限的惟一性、局部保号性与局部保序性。 8、下列运算有无错误?错在何处? 0 sIn (1) sIn lim r lim 9、设m,n∈N+,求下列极限 (1)Iim (1+ma)"-(1+na) (x+t) (x∈R); (4)lim sin 2r- sin 3x cos(a +h)-cos (7)lim √+x2+x)-(①1+x2-x)n (9)lim(1-r)tan- (10) lin e cos T (11) lim sin 5a-sin 3r 12)lim cos- cos 3.x SIn 2a (13)lim1-2x; (14)lim(1+ (15)lir (16)imn(cos=)(a≠0) cos(n arccos (n为奇数); (18)lim (cos vn+I-cos V (19) lim cos =cos (20) lim sin(x√n2+1) (21)lim(sin z) (22)lim sin -+cos +In 10、设{a,b是一个有限闭区间,如果vro∈[a,b,limf(x)存在且有限,证明:f(x)在[,b 上有界 11、设∫:(a,b)→R是无界函数,证明:存在数列{xn}c(a,b),使得limf(xn)=∞ 12、设f:园,+∞)→B,证明:、f(如)存在且有限台→v>0,3M>0使得Vx,z2>M 恒有|f(x1)-f(x2川<E3 6  (1) limx→∞ f(x) = a ⇐⇒ lim x→+∞ f(x) = lim x→−∞ f(x) = a ; (2) limx→x0 f(x) = a ⇐⇒ lim x→x+ 0 f(x) = lim x→x− 0 f(x) = a. 7  @A;<=>A;;<=BA# 8 2/CD$E?FÆ?#@Æ (1) limx→0 sin x x = limx→0 sin x limx→0 x = 0 0 = 1; (2) limx→∞ sin x x = limx→∞ sin x limx→∞ x = 0; (3) limx→0 x sin 1 x = limx→0 x limx→0 sin 1 x = 0. 9 " m, n ∈ N+ G2/  (1) limx→0 (1 + mx)n − (1 + nx)m x2 ; (2) limx→1  m 1 − xm − n 1 − xn  ; (3) limt→0 (x + t)n − xn t (x ∈ R); (4) limx→0 √ n 1 + x − 1 x ; (5) limx→0 sin 2x − sin 3x x ; (6) limh→0 cos (x + h) − cos x h ; (7) limx→0 ( √1 + x2 + x)n − ( √1 + x2 − x)n x ; (8) limx→0+ x  1 x ; (9) limx→1 (1 − x) tan πx 2 ; (10) limx→0 x2 1 − cos x ; (11) limx→0 sin 5x − sin 3x sin 2x ; (12) limx→0 cos x − cos 3x x . (13) limx→0 √x 1 − 2x; (14) limx→∞  1 + 2 x −x ; (15) limx→∞ x2 − 1 x2 + 1 x2 ; (16) lim x→+∞  cos a x x2 (a = 0); (17) limx→0 cos (n arccos x) x (nHI); (18) lim n→+∞  cos √ n + 1 − cos √n  ; (19) lim n→+∞ cos x 2 · cos x 22 ··· cos x 2n ; (20) lim n→+∞ sin (π n2 + 1); (21) lim x→π 2 (sin x) tan x; (22) limx→∞  sin 1 x + cos 1 x x ; (23) lim n→+∞ n + x n − 1 n ; (24) lim n→+∞ n + ln n n − ln n  n ln n . 10 " [a, b] !A$ BJCKD ∀x0 ∈ [a, b], limx→x0 f(x) 8#Æ$  f(x) # [a, b] L$E# 11 " f : (a, b) → R !EE8#/ {xn} ⊂ (a, b), MF limn→∞ f(xn) = ∞. 12 " f : [a, +∞) → R,  lim x→+∞ f(x) 8#Æ$ ⇐⇒ ∀ε > 0, ∃M > 0 MF ∀x1, x2 > M, G$ |f(x1) − f(x2)| < ε