(1-(cosx)^0.5)/(1-cos(x)^0.5)的极限
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(1-(cosx)^0.5)/(1-cos(x)^0.5)的极限
lim(x->0) [ 1- (cosx)^(1/2) ]/ [ 1- cos((x)^(1/2) ] (0/0)
=lim(x->0) [ (-1/2)(sinx)(cosx)^(-1/2) ]/ [ (1/2) x^(-1/2) sin((x)^(1/2)) ]
=lim(x->0) [ -x(cosx)^(-1/2) ]/ [ x^(-1/2) sin((x)^(1/2)) ] ( sinx~x)
=lim(x->0) [ -x^(3/2)(cosx)^(-1/2) ]/ sin((x)^(1/2)) (0/0)
=lim(x->0) - [(3/2)x^(1/2)(cosx)^(-1/2) -(1/2) x^(3/2).cosx.(sinx)^(-3/2) ]/ (1/2)x^(-1/2)cos((x)^(1/2))
=lim(x->0) - x^(1/2)[3x^(1/2)(cosx)^(-1/2) - x^(3/2).cosx.(sinx)^(-3/2) ]/ cos((x)^(1/2))
=0
=lim(x->0) [ (-1/2)(sinx)(cosx)^(-1/2) ]/ [ (1/2) x^(-1/2) sin((x)^(1/2)) ]
=lim(x->0) [ -x(cosx)^(-1/2) ]/ [ x^(-1/2) sin((x)^(1/2)) ] ( sinx~x)
=lim(x->0) [ -x^(3/2)(cosx)^(-1/2) ]/ sin((x)^(1/2)) (0/0)
=lim(x->0) - [(3/2)x^(1/2)(cosx)^(-1/2) -(1/2) x^(3/2).cosx.(sinx)^(-3/2) ]/ (1/2)x^(-1/2)cos((x)^(1/2))
=lim(x->0) - x^(1/2)[3x^(1/2)(cosx)^(-1/2) - x^(3/2).cosx.(sinx)^(-3/2) ]/ cos((x)^(1/2))
=0
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