lim (1+a/x)^(b x)
x->infinity
更新1:
lim (1+a/x)^(b x) x->infinity
evaluate limit by l'hospitals
回答 (2)
2009-11-10 11:16 pm
✔ 最佳答案
Let y = (1 + a/x)bx, then圖片參考：http://i388.photobucket.com/albums/oo325/loyitak1990/Nov09/Crazylim2.jpg
Hence, lim (x→∞) y = eab
參考: Myself
2009-11-10 11:21 pm
Let y=(1+a/x)^(bx), take ln, lny=bx(1+a/x)
lim (x->∞) lny
=lim (x->∞) (1+a/x)/(1/bx) [0/0 type]
=lim (x->∞) [(-a/x^2)/(1+a/x)]/[-1/(bx^2)]
=lim (x->∞) [ab/(1+a/x)]
=ab
So lim (x->∞) (1+a/x)^(b x) =e^[ab]
lim (x->∞) lny
=lim (x->∞) (1+a/x)/(1/bx) [0/0 type]
=lim (x->∞) [(-a/x^2)/(1+a/x)]/[-1/(bx^2)]
=lim (x->∞) [ab/(1+a/x)]
=ab
So lim (x->∞) (1+a/x)^(b x) =e^[ab]
收錄日期: 2021-04-26 13:50:16
原文連結 [永久失效]:
https://hk.answers.yahoo.com/question/index?qid=20091110000051KK00783