求极限 x趋向于π/3 ((1-2cosx)ln(1+x))/sin(x-π/3)
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求极限 x趋向于π/3 ((1-2cosx)ln(1+x))/sin(x-π/3)
![求极限 x趋向于π/3 ((1-2cosx)ln(1+x))/sin(x-π/3)](/uploads/image/z/17782560-0-0.jpg?t=%E6%B1%82%E6%9E%81%E9%99%90+x%E8%B6%8B%E5%90%91%E4%BA%8E%CF%80%2F3+%28%281-2cosx%29ln%281%2Bx%29%29%2Fsin%28x-%CF%80%2F3%29)
∵lim(x->π/3)[(1-2cosx)/sin(x-π/3)]
=lim(x->π/3)[2sinx/cos(x-π/3)] (0/0型极限,应用罗比达法则)
=2sin(π/3)/cos(π/3-π/3)
=2*(√3/2)
=√3
∴lim(x->π/3){[(1-2cosx)ln(1+x)]/sin(x-π/3)]
=lim(x->π/3)[(1-2cosx)/sin(x-π/3)]*lim(x->π/3)ln(1+x)
=√3*ln(1+π/3).
=lim(x->π/3)[2sinx/cos(x-π/3)] (0/0型极限,应用罗比达法则)
=2sin(π/3)/cos(π/3-π/3)
=2*(√3/2)
=√3
∴lim(x->π/3){[(1-2cosx)ln(1+x)]/sin(x-π/3)]
=lim(x->π/3)[(1-2cosx)/sin(x-π/3)]*lim(x->π/3)ln(1+x)
=√3*ln(1+π/3).
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