求极限:x趋向于1,lim(m/1-x^m—n/1-x^n)
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求极限:x趋向于1,lim(m/1-x^m—n/1-x^n)
令: x = 1+t
1 - x^m = 1 -(1+t)^m = -[mt + m(m-1)/2*t^2 + o(t^2)]
1 - x^n = 1 -(1+t)^n = -[nt + n(n-1)/2*t^2 + o(t^2)]
lim(m/1-x^m—n/1-x^n)
=lim [m(1-x^n) - n(1-x^m)]/(1-x^m)(1-x^n)
=lim { -[mnt + mn(n-1)/2*t^2 + o(t^2)] + [mnt + nm(m-1)/2*t^2 + o(t^2)]/(nmt^2 + o(t^2))
= mn(m-n)/2
1 - x^m = 1 -(1+t)^m = -[mt + m(m-1)/2*t^2 + o(t^2)]
1 - x^n = 1 -(1+t)^n = -[nt + n(n-1)/2*t^2 + o(t^2)]
lim(m/1-x^m—n/1-x^n)
=lim [m(1-x^n) - n(1-x^m)]/(1-x^m)(1-x^n)
=lim { -[mnt + mn(n-1)/2*t^2 + o(t^2)] + [mnt + nm(m-1)/2*t^2 + o(t^2)]/(nmt^2 + o(t^2))
= mn(m-n)/2
求极限:x趋向于1,lim(m/1-x^m—n/1-x^n)
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