Question

设f(x)在x=0的某领域内二阶导,且lim(sinx/x^3+f(x)/x^2)=0 求f(0)

设f(x)在x=0的某领域内二阶导,且lim(sinx/x^3+f(x)/x^2)=0 求f(0)求f(0) f'(0) f''(0) 及lim(x趋于0)(f(x)＋3)/x^2

Answer 1

用麦克劳林展开式来做
lim(x->0) [sin3x/x^3+f(x)/x^2]
=lim(x->0) [sin3x+xf(x)]/x^3
=lim(x->0) [3x-(3x)^3/6+o(x^3)+f(0)x+f'(0)x^2+f''(0)x^3/2+o(x^3)]/x^3
=lim(x->0) {[f(0)+3]x+f'(0)x^2+[f''(0)/2-9/2]x^3+o(x^3)}/x^3
=0
所以f(0)+3=0，f'(0)=0，f''(0)/2-9/2=0

即f(0)=-3，f'(0)=0，f''(0)=9
lim(x->0) [f(x)+3]/x^2
=lim(x->0) f'(x)/2x
=lim(x->0) f''(x)/2
=9/2