Question

求极限 具体过程

Answer 1

x->0

分母

tanx= x+(1/3)x^3+o(x^3)

sinx= x-(1/6)x^3+o(x^3)

tanx -sinx = (1/2)x^3 +o(x^3)

分子

tanx =x+(1/3)x^3+o(x^3)

tan(tanx)

=tan[x+(1/3)x^3+o(x^3)]

= [x+(1/3)x^3] +(1/3)[x+(1/3)x^3]^3 +o(x^3)

= [x+(1/3)x^3] +(1/3)[x^3+o(x^3)] +o(x^3)

=x +(2/3)x^3 +o(x^3)

sinx =  x-(1/6)x^3+o(x^3)

sin(sinx)

=sin[ x-(1/6)x^3+o(x^3)] 

= [ x-(1/6)x^3] -(1/6) [ x-(1/6)x^3] ^3 +o(x^3)

= [ x-(1/6)x^3] -(1/6) [ x^3+o(x^3)] +o(x^3)

=x-(1/3)x^3 +o(x^3)

tan(tanx)- sin(sinx)

=[x +(2/3)x^3 +o(x^3)] -[x-(1/3)x^3 +o(x^3)]

= x^3 +o(x^3)

lim(x->0) [tan(tanx) -sin(sinx)]/(tanx-sinx)

=lim(x->0) x^3/[(1/2)x^3]

=2