高等数学A上册习题解答 至微分中值定理(5)

华中师范大学专用

14. 0<a<b§¼êf(x)3[a,b]þëY§3(a,b)þ §Á|^…Ü¥

b

½ny²§ 3 :ξ∈(a,b)§¦f(b) f(a)=ξf (ξ)lny²µ-F(x)=lnx,x∈[a,b].K§F (x)=1=0,x∈[a,b].¦^…Ü¥

½n § 3 :ξ∈(a,b),¦

f(b) f(a)f (F(b) F(a)=ξ)

F

(ξ)

=

f(b) f(a)=ξf (ξ)ln

b

a

15.|^¼ê Vúª¦e 4 µ

x2

(1)xlimcosx e →∞

(2)xlim

sinx xcosx

→0

(1)):

cosx e x2

xlim

→∞x[x+ln(1 x)]x24x2x4=lim

[1

+

x+o(x4)] [1

+

+o(x4)]

x→∞

x2[x x

x+o(x2)]

x4

4

=lim

x+o(x4)

x→∞

x4

+o(x4)

=16

(2)):

lim

sinx xcosx

x→0tanx

x2=[x x3

+o(x3

)] x[1 +o(x2)]

xlim→0x3

=xlim x3

+x3+o(x3

)]→0x3=15

(0.4)

(0.5)