(2) lim In(tan x)=In(tan)=0 2 3)im(√x2+2-√x2-x)=lim 2+x lim (4)imyx+2-2 x→2(x-2)√x+2+2) x+2+2 x sin (5) lim x (6) lim x(/1+--1)=lim x( Im √x +1+√x 6.计算下列极限: (1)lim tan 2x (2) lim e (3) lim(1+2 x (5) lim n[In(1+n)-Innl 解(1)mtan2x=mx=2 x→0x 2x+1 (2)lime f =er =e=l (3)lim(1+ 2 tan x) x=[lim(1+2 tan x)2tan']=e (4)im lim(3 (2) π 4 π lim ln(tan ) ln(tan ) 0 x 4 x → = = . (3) 2 2 2 2 2 2 1 2 lim ( 2 ) lim lim 2 2 1 1 1 x xx x x x xx x xx x x →+∞ →+∞ →+∞ + + +− − = = ++ − + + − 1 2 = . (4) 22 2 22 2 1 1 lim lim lim xx x 2 4 ( 2)( 2 2) 2 2 x x →→ → x xx x +− − = == − − ++ ++ . (5) 2 2 000 1 1 sin sin 1 lim lim lim sin 0 xxx sin 2 2 2 x x x x x →→→ xx x = == . (6) 1 1 lim ( 1 1) lim ( 1) lim ( 1 ) xxx x x x xx x →+∞ →+∞ →+∞ x x + + − = − = +− 1 1 lim lim 1 1 2 1 1 x x x x x x →+∞ →+∞ = == + + + + . 6. 计算下列极限: (1) 0 tan 2 lim x x → x ; (2) 2 2 1 lim e x x x + →∞ ; (3) 2 2 cot 0 lim(1 2 tan ) x x x → + ; (4) 2 2 2 1 lim ( ) 1 x x x →∞ x − + ; (5) lim [ln(1 ) ln ] n n nn →∞ + − ; (6) 1 1 cot lim ( ) x x x →∞ x − . 解 (1) 0 0 tan 2 2 lim lim 2 x x x x → → x x = = . (2) 2 2 21 21 lim 0 lim e e e 1 x x x x x x →∞ + + →∞ = == . (3) 2 2 1 2 cot 2 2 2 2tan 0 0 lim(1 2 tan ) [lim(1 2 tan ) ] e x x x x x x → → + =+ = . (4) 22 2 2 2 2 2 12 2 lim 2 1 1 2 2 2 1 2 lim ( ) lim (1 ) e e 1 1 x xx x x x x x x x x x →∞ +− − − ⋅ + + − →∞ →∞ − =− = = + +