(1)
x->0
e^(sinx)
=1+(sinx)+(1/2)(sinx)^2+(1/6)(sinx)^3+ (1/24)(sinx)^4 +o(x^4)
=1+[x-(1/6)x^3 +o(x^4) ]+(1/2)[x-(1/6)x^3 +o(x^4) ]^2
+(1/6)[x^3+o(x^4)]+ (1/24)[x^4+o(x^4)] +o(x^4)
=1+[x-(1/6)x^3 +o(x^4) ]+(1/2)[x^2-(1/3)x^4 +o(x^4) ]
+(1/6)[x^3+o(x^4)]+ (1/24)[x^4+o(x^4)] +o(x^4)
=1+x +(1/2)x^2 + ( -1/6+1/24)x^4 +o(x^4)
=1+x +(1/2)x^2 -(1/8)x^4 +o(x^4)
x+e^(sinx)
=1+2x +(1/2)x^2 -(1/8)x^4 +o(x^4)
=1+x[2 +(1/2)x-(1/8)x^3 +o(x^3)]
( x+ e^(sinx))^(1/x)
={ 1+x[2 +(1/2)x -(1/8)x^3 +o(x^3)] }^(1/x)
=e^[2 +(1/2)x -(1/8)x^3 +o(x^3)]
同样地
sinx+ e^x
= 1+ 2x +(1/2)x^2 +(1/24)x^4 +o(x^4)
=1 +x[2 +(1/2)x +(1/24)x^3 +o(x^3)]
( sinx+ e^x )^(1/x)
={1 +x[2 +(1/2)x +(1/24)x^3 +o(x^3)] }^(1/x)
=e^[2 +(1/2)x +(1/24)x^3 +o(x^3)]
//
lim(x->0) [( x+ e^(sinx))^(1/x) -( sinx+ e^x )^(1/x)]/x^3
=lim(x->0) { e^[2 +(1/2)x -(1/8)x^3 +o(x^3)]-e^[2 +(1/2)x +(1/24)x^3 +o(x^3)] }/x^3
抽出共同因子 e^[2 +(1/2)x +o(x^3)]
=lim(x->0) { e^[2 +(1/2)x+o(x^3)] } .{ e^[-(5/24)x^3]-e^[(1/24)x^3] }/x^3
lim(x->0) e^[2+(1/2)x+o(x^3)]=e^2
=e^2.lim(x->0) { e^[-(1/8)x^3]-e^[(1/24)x^3] }/x^3
等价无穷小
=e^2.lim(x->0) [-(1/8)x^3- (1/24)x^3] /x^3
=e^2.lim(x->0) -(1/6)x^3 /x^3
=-(1/6)e^2
(2)
x->0
ln(1+3x) = 3x -(1/2)(3x)^2 +o(x^2)=3x -(9/2)x^2 +o(x^2)
f(x) = f(0) +f'(0)x +o(x)
xf(x) = xf(0) +f'(0)x^2 +o(x^2)
ln(1+3x) +xf(x)
=[3+f(0)]x +[-(9/2)+f'(0)]x^2 +o(x^2)
lim(x->0) [ln(1+3x) +xf(x) ]/x^2 =-1
=>
3+f(0)=0 and -(9/2)+f'(0) =-1
f(0) =-3 and f'(0) =7/2
f(x) =-3 +(7/2)x +o(x)
lim(x->0) (3+f(x))/x
=lim(x->0) (7/2)x/x
=7/2
