Homework Help

Question about Calculus (Limit)...?In my Textbook, the question is: Show that : lim...

user profile pic

wikkibahi | Student, Undergraduate | (Level 3) eNoter

Posted December 1, 2011 at 2:33 PM via web

dislike 0 like

Question about Calculus (Limit)...?

In my Textbook, the question is:

Show that : lim (1+1/x)^x= e  as  x-->  Infinity

1 Answer | Add Yours

user profile pic

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted December 1, 2011 at 11:11 PM (Answer #1)

dislike 0 like

`Put 1 + 1/x = t =gt 1/x = t - 1 =gt x = 1/(t-1)`

Calculate logarithms both sides:

`ln x = ln(1/(t-1))`

`` Use quotient property of logarithms:

ln x = ln1 - ln (t-1)

ln 1 = 0 =>

`ln x = - ln (t-1)`

Write the function using variable t:

t^(1/(t-1)) => `lim_(t-gtoo)`  `t^(1/(t-1))`  = `lim_(t->oo)` `(1 + t - 1)^(1/(t-1)) ` (you must add and subtract 1 in the brackets and create the form `(1+a_n)^(1/a_n)).` Calculate the limit:

 `lim_(t->oo)`  [1 + (t-1)]^1/(t-1) = e

Join to answer this question

Join a community of thousands of dedicated teachers and students.

Join eNotes