lim(x→0)(1+3tan^2(x))^cot^2(x)=?
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lim(x→0)(1+3tan^2(x))^cot^2(x)=?
lim(x→0)(1+3tan^2(x))^cot^2(x)=?
lim(x→0)(1+3tan^2(x))^cot^2(x)=?
楼上的是取指数,再取对数,其实这个题目就是两个重要极限:lim(x→∞)(1+1/x)^x=e
注意到tanx和cotx互为倒数,因此
lim(x→0)(1+3tan^2(x))^cot^2(x)
=lim(x→0)(1+3tan^2(x))^[1/tan^2(x)]
=lim(x→0)(1+3tan^2(x))^[3/(3tan^2(x))]
=lim(x→0)(1+3x^2)^[3/(3x^2)]
=e^3
lim(x→0)(1+3tan^2(x))^cot^2(x)
=e^[lim(x→0)(3tan^2(x)*cot^2(x)]
=e^3
20.09
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