高数！求详细过程！
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(x→+∞) lim (π/2-arctanx)^(1/lnx)
令t=arctanx，则x=tant
(t→π/2) lim (π/2-t)^(1/lntant)
= e^(t→π/2) lim ln(π/2-t) / lntant ....[0/0]
= e^(t→π/2) lim [-1/(π/2-t)] / [1/tant * 1/cos²t]
= e^(t→π/2) lim [-1/(π/2-t)] / [1/(sintcost)]
= e^(t→π/2) lim (1/2)[sin2t/(t-π/2)] ....[0/0]
= e^(t→π/2) lim (1/2) [2cos2t / 1]
= e^(t→π/2) lim cos2t
= e^(-1)
= 1/e
令t=arctanx，则x=tant
(t→π/2) lim (π/2-t)^(1/lntant)
= e^(t→π/2) lim ln(π/2-t) / lntant ....[0/0]
= e^(t→π/2) lim [-1/(π/2-t)] / [1/tant * 1/cos²t]
= e^(t→π/2) lim [-1/(π/2-t)] / [1/(sintcost)]
= e^(t→π/2) lim (1/2)[sin2t/(t-π/2)] ....[0/0]
= e^(t→π/2) lim (1/2) [2cos2t / 1]
= e^(t→π/2) lim cos2t
= e^(-1)
= 1/e
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详细过程如图rt所示
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