南京大学微积分综合题一.pdf

?.? 1. lim n sin2 ?n2 + n?, 2. lim n ?cos x n + sin x n ?n ? 3. lim x0 (cosx) 1 sin2x ? 4. lim x0+ xx? 5. lim x0 ?ax + bx+ cx 3 ?1 x ? 6. lim x0 ?tanx x ?Cot2x ? 7. lim x0 3 p 1 + sin2x ex 2 x2 ? 8. lim x0+ xsin x 1 xx 1 ? 9. lim n n3 “ k n2 k X i=1 1 (n + i)2 # (k N)? 10. lim n n2( n a n+1 a) (a 0) ? 11. lim x0 exsinxx(1+x) sin3x ? 12.?f(x)?x = a?f(a) 6= 0?lim n ?f(a + 1 n) f(a) ?n ? 13.?lim x1 ? a 1x2 x 1x ? = 3 2 ?a? 14.?esinx etanx?x? ?.? 1.?f(x) = lim n 2nx+ 1 2nx 1 ?f(x)? 2.?f(x) = lim n n r 1 + 1 xn + (lnx)n+ xn 1 + xn ! ?f(x)? ? 1 3.?f(x) = 1 2 8 1 x ? ?.? 1.?f(x)?0,1?f(0) = 0?f(1) = 1?Æ? (0,1)? f( 1 3) = f() 1 3. 2.?f(x)?0,1?f(0) = 0?max x0,1 f(x) = 1?Æ?c (0,1)? f(c) = c. 3.?f(x)?0,2a?f(0) = f(2a)?Æ? 0,a?f() = f( + a)? ?.? 1.?y = ax2?y = x?a? ? 2.?y = tannx?( 4,1) ?x?(n,0)?lim n y(n)? 3.?y = f(x) = anx(a 0, a 6= 1?n?)?(0,1)?x? (xn,0)?lim n(1 + xn) n ? 4.?2x 6y + 1 = 0?y = x3 3x 5? ?.? 1.?lim x ? x2 x+1 ax b ? = 0?a?b? 2.?lim x + ? e 1 x p 1 + x2 ax b ? = 0?a,b? 3.?lim x0 ?1 + x ax?x2 = b?a,b?a,b? ?.? 1.?y = y(x)?arctan y x = ln px2 + y2? dy dx ? 2.? ( x = a(t sint) y = a(1 cost) ? dy dx ?d 2y dx2 ? 3.?y = f(x)? ( x = t + lnt y = 3t2+ 2t3 ? dy dx ?d 2y dx2 ? 2 ?.?Lebnitz? 1.?y = 1 x2x2 ?y(n)(n 1)? 2.?f(x) = x3sinx?f(10)(0)? 3.?y = x2lnx?y(n)? 4.?y = arcsinx?(1 x2)y= xy?y(n)(0)? 5.?y = 1 x2+x+2 ?y(n)? ?.? 1.?A,B? ? f(x) = ( sinx + 2Aex,x 0 ?f(x)?x = 0? 3.?f(x) = ( 1 x 1 ex1, x 6= 0 a,x = 0 ?a? ?f(x)?x = 0?f(x)? ?f(x)?x = 0? 4.?f(x) = ( 1 x(asinx + bcosx b), x 0 ?x = 0? (1).?a,b? ? (2).?f(0)? (3).?f(x)?x = 0? 4