Question

高等数学求极限问题？

划红线的部分怎么得出来的？书中没有解释

Answer 1

x->0

分子

√(1+x^2) = 1+ (1/2)x^2 +o(x^2)

x+√(1+x^2) = 1+ x+(1/2)x^2 +o(x^2)

ln[x+√(1+x^2)}

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

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

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

=x +o(x^2)

ln(1+x)  = x -(1/2)x^2 +o(x^2)

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

分母

ln[x+√(1+x^2)] = x +o(x^2)

ln(1+x) =x+o(x)

ln[x+√(1+x^2)] .ln(1+x) = x^2 +o(x^2)

lim(x->0+) f(x)

=lim(x->0+){ 1/ln[x+√(1+x^2)] - 1/ln(1+x) }

=lim(x->0+) { ln(1+x) -ln[x+√(1+x^2)] }/{ ln[x+√(1+x^2)].ln(1+x) }

=lim(x->0+)  -(1/2)x^2 / x^2

=-1/2