Question

22题求极限

麻烦详细一下谢谢了

Answer 1

(22)

分母

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

√(1+x^2) = 1+(1/2)x^2+o(x^2)

x+√(1+x^2) = 1+x +o(x)

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

分子

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

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

=x +o(x^2)

ln(1+x) -ln[x+√(1+x^2)] 

= [x-(1/2)x^2+o(x^2)]-[x +o(x^2)]

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

//

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) { ln(1+x) -ln[x+√(1+x^2)] }/x^2

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

=-1/2