Question

想问一下求极限用泰勒公式这么化简为什么不对？

Answer 1

x->0
分子
(sinx)^2 = x^2 +o(x^2)
e^x = 1+ x +(1/2)x^2 +o(x^2)
(sinx)^2 +e^x = 1+ x +(3/2)x^2 +o(x^2)
ln[(sinx)^2 + e^x]
=ln[1+ x +(3/2)x^2 +o(x^2)]
=[x +(3/2)x^2] -(1/2)[x +(3/2)x^2]^2 +o(x^2)
=[x +(3/2)x^2] -(1/2)[x^2 +o(x^2)] +o(x^2)
=x + x^2 +o(x^2)
ln[(sinx)^2 + e^x] -x = x^2 +o(x^2)
分母

e^(2x) = 1+ 2x + 2x^2 +o(x^2)
(sinx)^2 + e^x = 1+ 2x + 3x^2 +o(x^2)
ln[x^2+e^(2x)]
=ln[1+ 2x + 3x^2 +o(x^2)]
=[2x + 3x^2 ] -(1/2)[2x + 3x^2 ]^2 +o(x^2)
=[2x + 3x^2 ] -(1/2)[4x^2+o(x^2)] +o(x^2)
=2x + x^2 +o(x^2)
ln[x^2+e^(2x)] -2x = x^2 +o(x^2)

/
lim(x->0) {ln[ (sinx)^2 + e^x ] -x }/{ ln[x^2+e^(2x)] -2x }

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

Answer 2

是对的，希望有所帮助