计算lim(x趋向于π/2)(sinx)的tanx次方怎么计算?
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计算lim(x趋向于π/2)(sinx)的tanx次方怎么计算?
计算lim(x趋向于π/2)(sinx)的tanx次方怎么计算?
计算lim(x趋向于π/2)(sinx)的tanx次方怎么计算?
解法一：∵lim(x->π/2)[(sinx-1)tanx]
=lim(x->π/2){[(sinx-1)/cosx]sinx}
=lim(x->π/2)[(sinx-1)/cosx]*lim(x->π/2)(sinx)
=lim(x->π/2){[sin(x/2)-cos(x/2)]/[cos(x/2)+sin(x/2)]}*1
=0*1
=0
lim(x->π/2){(sinx)^[1/(sinx-1)]}
=lim(x->π/2){(1+sinx-1)^[1/(sinx-1)]}
=e (应用特殊极限lim(x->0)[(1+x)^(1/x)]=e)
∴原式=lim(x->π/2)[(sinx)^tanx]
=lim(x->π/2)【(sinx)^{[1/(sinx-1)]*[(sinx-1)tanx]}】
=【lim(x->π/2){(sinx)^[1/(sinx-1)]}】^{lim(x->π/2）[(sinx-1)tanx]}
=e^{lim(x->π/2)[(sinx-1)tanx]}
=e^0
=1.
解法二：原式=lim(x->π/2)[(sinx)^tanx]
=lim(x->π/2){e^[tanx*ln(sinx)]}
=e^{lim(x->π/2)[tanx*ln(sinx)]}
=e^{lim(x->π/2)[ln(sinx)/cotx]}
=e^[lim(x->π/2)（-cotx/csc²x)]
=e^[lim(x->π/2)（-sinx*cosx)]
=e^0
=1.
取自然对数得
lim(x→π/2）tanxlnsinx
=lim(x→π/2）lnsinx/cotx
=lim(x→π/2)(cosx/sinx)/(-csc^2x)
=lim(x→π/2)(-sinxcosx)
=0
因此
lim(x→π/2）（sinx）＾tanx
=lim(x→π/2）e^(tanxlnsinx)
=1