x趋于0 求（ln(1＋x²)－ln(1＋sin²x)）/xsin³x的极限

题目

x趋于0 求（ln(1＋x²)－ln(1＋sin²x)）/xsin³x的极限

答案

用等价无穷小代换.分子ln(1+x^2)-ln(1+sin(x)^2)= ln((1+x^2)/(1+sin(x)^2))= ln(1+(x^2-sin(x)^2)/(1+sin(x)^2)),与(x^2-sin(x)^2)/(1+sin(x)^2)是等价无穷小(因为ln(1+t) t),即与x^2-sin(x)^2是等价无穷小(因为1+s...