lim(x趋于π/2)(tanx/tan3x)求极限

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lim(x趋于π/2)(tanx/tan3x)求极限

lim(x趋于π/2)(tanx/tan3x)求极限
lim(x趋于π/2)(tanx/tan3x)求极限

lim(x趋于π/2)(tanx/tan3x)求极限
lim(x→π/2)tan3x/tanx
=lim(x→π/2) 3sec^2(3x)/sec^2x
=lim(x→π/2) 3cos^2x/cosx^2(3x)
=lim(y→0) 3sin^2y/sin^2(3y) (令y=π/2-x)
=lim(y→0) 3y^2/(3y)^2
=3/9
=1/3

三分之一

lim(x趋于π/2)(tanx/tan3x)=lim(x趋于0)(cotx/cot3x)=lim(x趋于0)(tan3x/tanx)=3

tanx/tan3x=(sinx/cosx)/(sin3x/cos3x)
=-sinx/sin3x=-1/3

3