Question

泰勒公式求极限。

Answer 1

根据题意，sin6x-tanx*f(x)~o(x^3)
根据泰勒展开，sin6x=6x-(6x)^3/3!+o(x^4)，tanx=x+x^3/3+o(x^4)
f(x)=f(0)+f'(0)*x+f''(0)/2!*x^2+o(x^2)

所以sin6x-tanx*f(x)
=[6x-(6x)^3/3!+o(x^4)]-[x+x^3/3+o(x^4)][f(0)+f'(0)*x+f''(0)/2*x^2+o(x^2)]
=[6-f(0)]x-f'(0)*x^2-[36+f(0)/3+f''(0)/2]*x^3+o(x^3)
~o(x^3)
所以6-f(0)=0，且f'(0)=0，且36+f(0)/3+f''(0)/2=0
f(0)=6，f'(0)=0，f''(0)=-76
所以f(x)=6-38x^2+o(x^2)
lim(x->0) [6-f(x)]/x^2
=lim(x->0) [6-6+38x^2-o(x^2)]/x^2
=38