Question

请教大神求超详细步骤解析！♥ 1.求极限limx趋向于0√2+sinx(sinx-x)/tan³x

请教大神，求详细步骤解析！♥
1.求极限limx趋向于0√2+sinx(sinx-x)/tan³x
2.求极限limx趋向于+∞〔x²（e的1/x次方-1）-x〕

Answer 1

(1)
根据泰勒展式
f(x) = f(0) +[f'(0)/1!]x +[f''(0)/2!]x^2+....+[f^(n)(0)/n!]x^n +....
sinx = x- (1/6)x^3 +o(x^3)
x->0
sinx= x-(1/6)x^3 +o(x^3)
sinx -x =(1/6)x^3 +o(x^3)
lim(x->0) √(2+sinx) (sinx-x)/ (tanx)^3
=√2.lim(x->0) (sinx-x)/ (tanx)^3
=√2.lim(x->0) (sinx-x)/ x^3
=√2.lim(x->0) (1/6)x^3/ x^3
=√2/6
(2)
let
u=1/x
lim(x->∞) [ x^2. (e^(1/x) -1 ) -x ]
=lim(u->0) [ (1/u)^2. (e^u -1 ) -1/u ]
=lim(u->0) [(e^u -1 ) -u]/u^2