lim(1+(3/x))^x

👨🏻‍🏫 學術台數學

[匿名]

2012-08-20 12:42 am

i want to ask how to solve the problem....

lim<x to infinity>(1+(3/x))^x (ans: e^3)

why the ans invole e?

why not calculated as follow?
(lim1 + 3lim(1/x))^x
=(1+3(0))^x
=1??

thx

更新1:

讀文學的學生: but why the following method is not correct? (lim1 + 3lim(1/x))^x =(1+3(0))^x =1??

回答 (2)

PeterChow

2012-08-20 2:49 am

✔ 最佳答案

lim(x→∞)[1+(3/x)]^x
=lim(x→∞){[1+1/(x/3)]^(x/3)}^3
=e^3
Remember,
lim(x→∞)[(1+1/n)^n]=e

參考: me

👍 0 👎 0

[匿名]

2012-08-20 1:26 am

記唔記得呢條式:
lim(n-->infinity)(1+x/n)^n = e^x = 1+x+(x^2)/2! + (x^3)/3! + ........

咁對番D位,個答案就係e^3啦

2012-08-19 20:22:20 補充：
x係變數,你唔可以唔理個x次方而張limit擺入(1+3/x)

你咁樣做只係求左個y既limit,之後先再張舊數放番入X次方,咁樣既話就唔係求成舊數既limit

2012-08-19 20:29:23 補充：
個y係指番個(1+3/x)

參考: me, me, me

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