Question

高数 极限 求解

Answer 1

x->0

分子：

√(1+x^2) = 1+(1/2)x^2-(1/8)x^4 +o(x^4)

1+(1/2)x^2 -√(1+x^2) =(1/8)x^4 +o(x^4)

分母

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

e^(x^2)= 1+x^2 +o(x^2)

cosx -e^(x^2) = -(3/2)x^2 +o(x^2)

sin(x^2) = x^2 +o(x^2)

[cosx -e^(x^2)].sin(x^2)

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

=-(3/2)x^4 +o(x^4)

lim(x->0) [1+(1/2)x^2 -√(1+x^2)]/{ [cosx -e^(x^2)].sin(x^2) }

=lim(x->0) (1/8)x^4/[-(3/2)x^4]

=-1/12