51问答网 > x趋于0时(1+tanx)^1⼀2-(1+sinx)^1⼀2⼀x(1+sinx^2)-x等于

x趋于0时(1+tanx)^1⼀2-(1+sinx)^1⼀2⼀x(1+sinx^2)-x等于

x趋于0时(1+tanx)^1/2-(1+sinx)^1/2/x(1+sinx^2)-x等于

2025-03-10 00:48:54

推荐回答（1个）

回答1：

x->0
tanx ~ x +(1/3)x^3
sinx ~ x-(1/6)x^3
tanx -sinx ~ (1/2)x^3
----------------
x(1+sinx^2)-x ~ x(1+x^2) -x ~ x^3
------------------------------
lim(x->0) [(1+tanx)^(1/2)-(1+sinx)^(1/2) ]/[ x(1+sinx^2)-x]
=lim(x->0) [(1+tanx)-(1+sinx)]/{ [ x(1+sinx^2)-x] .[(1+tanx)^(1/2)+(1+sinx)^(1/2) ] }
=(1/2) lim(x->0) (tanx-sinx)/ [ x(1+sinx^2)-x]
=(1/2) lim(x->0) (1/2)x^3/ x^3
=1/4