- 求极限lim(n→∞)tan^n(π/4+2/n)谢谢！我看不懂
- 我看不懂求解过程，请详细，谢谢！
- 求极限 lim(n→∞) tan^n (π/4 + 2/n)
解 lim(n→∞)tan^n(π/4+2/n)
=lim(n→∞)[(tan(π/4)+tan(2/n))/(1-tan(π/4)tan(2/n))]^n
=lim(n→∞)[(1+tan(2/n))/(1-tan(2/n))]^n
=lim(n→∞)(1+tan(2/n))^n/(1-tan(2/n))^n (1)
因为 lim(n→∞)(1+tan(2/n))^n
=lim(n→∞){[1+tan(2/n)]^(1/(tan(2/n))}^[2(tan(2/n)/(2/n)]
=e^2, (2)
lim(n→∞)(1-tan(2/n))^n
=lim(n→∞){[1-tan(2/n)]^(-1/(tan(2/n))}^[-2(tan(2/n)/(2/n)]
=e^(-2), (3)
由(1),(2),(3)得
lim(n→∞)tan^n(π/4+2/n)=e^2/e^(-2)=e^4.