lim(x趋向于无穷大时)x^2[1-cos(1/x)]

题目

lim(x趋向于无穷大时)x^2[1-cos(1/x)]

答案

lim x->∞ x²[1 - cos(1/x)]
= lim x->∞ [1 - cos(1/x)] / (1/x²) (分子, 分母同时趋向於0, 可利用洛必达法则)
= lim x->∞ {0 - [-sin(1/x)] * [-x^(-2)]} / [-2x^(-3)]
= lim x->∞ [sin(1/x)] / (2/x) (分子, 分母同时趋向於0, 可利用洛必达法则)
= lim x->∞ [cos(1/x) * (1/x)'] / (2/x)'
= lim x->∞ [cos(1/x)] / 2
= cos(0) / 2
= 1/2