Question

高数问题啊啊啊啊

limx→∞ x^2(1- xsin1/x)=?

Answer 1

您好！
解：lim<x→+∞>x²(1-xsin1/x)(令x=1/t)
=lim<t→0+>[1-(sint/t)]/t²
=lim<t→0+>(t-sint)/t³(洛必达法则)
=lim<t→0+>(1-cost)/3t²
=(1/3)lim<t→0+>(1-cost)/t²(等价无穷小代换)
=(1/3)(1/2)
=1/6

Answer 2

用洛必达法则
原式=lim(1-xsin(1/x))/(1/x²)
=lim(1/xcos(1/x)-sin(1/x))/(-2/x³)(利用洛必达法则)
=lim(-1/x²cos(1/x)+1/x³sin(1/x)+1/x²cos(1/x))/(6/x^4)(再次利用洛必达法则)
=lim(1/x³sin(1/x))/(6/x^4)
=1/6limsin(1/x)/(1/x)
=1/6