求下列极限，如图 10
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x->0
ln(1+x^2) = x^2 -(1/2)x^4 +o(x^4)
tanx = x+(1/3)x^3+o(x^3)
(tanx)^2 =[x+(1/3)x^3+o(x^3)]^2 = x^2 +(2/3)x^4+o(x^4)
ln(1+(tanx)^2)
=ln( 1+x^2 +(2/3)x^4+o(x^4) )
= (x^2 +(2/3)x^4+o(x^4)) -(1/2)[x^2 +(2/3)x^4+o(x^4)]^2 +o(x^4)
= (x^2 +(2/3)x^4+o(x^4)) -(1/2)[x^4+o(x^4)] +o(x^4)
=x^2 - (1/6)x^4 +o(x^4)
ln(1+x^2) -ln(1+(tanx)^2)
=[ x^2 -(1/2)x^4 +o(x^4)] -[x^2 - (1/6)x^4 +o(x^4)]
=-(1/3)x^4 +o(x^4)
lim(x->0) [ 1/ln(1+(tanx)^2) - 1/ln(1+x^2) ]
=lim(x->0) [ln(1+x^2) -ln(1+(tanx)^2) ] /[ln(1+(tanx)^2).ln(1+x^2) ]
=lim(x->0) [ln(1+x^2) -ln(1+(tanx)^2) ] / x^4
=lim(x->0) -(1/3)x^4 / x^4
=-1/3
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