x→1
lim (1-x)*tan(πx/2)
换元,t=1-x
=lim(t→0) t*tan(π(1-t)/2)
=lim t*tan(π/2-π*t/2)
=lim t*cot(π*t/2)
=lim t/tan(π*t/2)
=lim t/sin(π*t/2) * cos(π*t/2)
=(2/π)*lim (π*t/2)/sin(π*t/2) * lim cos(π*t/2)
根据重要的极限:lim sinx/x=1
=(2/π)*1*1
=2/π
有不懂欢迎追问
x→1
lim (1-x)*tan(πx/2)
换元,t=1-x
=lim(t→0) t*tan(π(1-t)/2)
=lim t*tan(π/2-π*t/2)
=lim t*cot(π*t/2)
=lim t/tan(π*t/2)
=lim t/sin(π*t/2) * cos(π*t/2)
=(2/π)*lim (π*t/2)/sin(π*t/2) * lim cos(π*t/2)
根据重要的极限:lim sinx/x=1
=(2/π)*1*1
=2/π
有不懂欢迎追问