Question

用泰勒公式证明极限题目！！！

lim（x→0）＝（e的x次方-sinx-1)/[1-根号下（1-x*x)]

Answer 1

e^x＝1＋x＋x^2/2＋o(x^2)
sinx＝x＋o(x^2)
所有，e^x－sinx－1＝1/2×x^2＋o(x^2)
√(1-x^2)＝1－1/2×x^2＋o(x^2)，所以1－√(1-x^2)＝1/2×x^2＋o(x^2)

lim(x→0) [e^x－sinx－1]/[1－√(1-x^2)]
＝lim(x→0) [1/2×x^2＋o(x^2)]/[1/2×x^2＋o(x^2)]
＝lim(x→0) [1/2＋o(x^2)/x^2]/[1/2＋o(x^2)/x^2]
＝[1/2＋0]/[1/2＋0]＝1

Answer 2

=lim[1+x+(1/2)x^2+(1/6)x^3+……-x+(1/6)x^3+……-1]/[1-1+(1/2)x^2+……]
=lim[(1/2)x^2+o(x^2)]/[(1/2)x^2+o(x^2)]
=lim(1/2+o(x^2)/x^2]/[1/2+o(x^2)/x^2]
=(1/2+0)/(1/2+0)
=1

Answer 3

没有分谁干呀？