Limit URGENT!~~

1. lim(x->0)_[ (1+x)^(1/2)－(1－x)^(1/2)]/[(1+x)^(1/3)－(1－x)^(1/3)]Sol
lim(x->0)_[ (1+x)^(1/2)－(1－x)^(1/2)]/[(1+x)^(1/3)－(1－x)^(1/3)]
=lim(x->0)_[(1/2)(1+x)^(－1/2)+(1/2)(1－x)^(－1/2)]
/[(1/3)(1+x)^(－2/3)+(1/3)(1－x)^(－2/3)]
=1/(2/3)
=3/2

2.y=x+(1/x^2)=(x^3+1)/x^2
dy/dx
=lim(h->0)_｛[(x+h)^3+1]/(x+h)^2]－(x^3+1)/x^2｝/h
=lim(h->0)_{[x^2(x+h)^3+x^2]－[(x+h)^2(x^3+1)]/[x^2(x+h)^2｝/h=lim(h->0)_{[x^2(x^3+3hx^2+3h^2x+h^3)+x^2]－[(x^2+2xh+h^2)(x^3+1)]
/[x^2(x+h)^2｝/h
=lim(h->0)_{[(x^5+3hx^4+3h^2x^3+h^3x^2+x^2)－
(x^5+x^2+2hx^4+2hx+h^2x^3+h^2)]/[x^2(x+h)^2]｝
=lim(h->0)_{(hx^4+2h^2x^3+h^3x^2－2hx－h^2)/[x^2(x+h)^2]｝/h
=lim(h->0)_{(x^4+2hx^3+h^2x^2－2x－h)/[x^2(x+h)^2]｝
=(x^4－2x)/[x^2(x)^2]
=1－2x^(－3)




2011-04-24 21:49:56 補充：
d[ (1+x)^(1/2)－(1－x)^(1/2)]/dx
=d (1+x)^(1/2)/dx－d(1－x)^(1/2)]/dx
=[d (1+x)^(1/2)/d(1+x)]*[d(1+x)/dx]－[d(1－x)^(1/2)/d(1－x)]*[d(1－x)/dx]]
=(1/2)(1+x)^(－1/2)－(1/2)(1－x)^(－1/2)*(－1)
=(1/2)(1+x)^(－1/2)+(1/2)(1－x)^(－1/2)

看不懂這個步驟q.1 =lim(x->0)_[(1/2)(1+x)^(－1/2)+(1/2)(1－x)^(－1/2)]
/[(1/3)(1+x)^(－2/3)+(1/3)(1－x)^(－2/3)]
可否解釋一下 thanks a lot!~