Search
Ask a question
0

What is the limit of this question?

Lim x-->∞  (1+1/x)x

4 Answers by Expert Tutors

Tutors, sign in to answer this question.
Andre W. | Friendly tutor for ALL math and physics coursesFriendly tutor for ALL math and physics ...
5.0 5.0 (3 lesson ratings) (3)
1
Hi David,
 
The limit lim x→∞ (1+1/x)x is of the indeterminate form 1. Before we can use l'Hospital's Rule, we need to bring it into the standard indeterminate form 0/0. For this we use the fact that
lim (ln f(x)) = ln (lim f(x))
So, consider
y = ln (1+1/x)x = x ln (1+1/x) = ln(1+1/x) / (1/x)
Then
limx→∞ y = limx→∞ ln(1+1/x) / (1/x)
which is of the form 0/0, so by l'Hospital's Rule,
limx→∞ y = lim (1/(1+1/x))(-1/x²) / (-1/x²) = lim 1/(1+1/x) = 1.
Therefore,
ln (lim x→∞ (1+1/x)x) = 1
lim x→∞ (1+1/x)x = e.
 
Note: this is how you get the continuous compound interest formula!
 
Brad M. | Professional Tutor: STEM plus Business, Accounting, and InvestmentProfessional Tutor: STEM plus Business,...
4.9 4.9 (219 lesson ratings) (219)
1
Hey David -- we may stairstep mentally ... (1+ 1/1)^1 = 2 ... (1.5)^2 = 2.25 ... 
(5/4)^4 = 625/256 ~ 2.5 ... "rule of 72" for doubling is nice ... (11/10)^10 = (1.10)^10
=> 2 at 7th, 2.2, 2.42, 2.68 ... (1.05)^20 => 2 at 14th, 2.1, 2.2, 2.31, 2.43, 2.55, 2.68
pretty nice convergence to 2.7 or "e" ... Best regards, sir :)
Ralph L. | Algebra I, II, Visual Basic, Beginning C++ tutorAlgebra I, II, Visual Basic, Beginning C...
4.0 4.0 (1 lesson ratings) (1)
0
limit of (1 + 1/x)x as x ---> ∞ is e.  e ≈ 2.71828........

Comments

try using a calculator. plug in as many 9's to x and you should get e.
 
Vivian L. | Microsoft Word/Excel/Outlook, essay composition, math; I LOVE TO TEACHMicrosoft Word/Excel/Outlook, essay comp...
3.0 3.0 (1 lesson ratings) (1)
-1
Hi David;
x cannot equal zero because zero cannot be a numerator.
Otherwise, there is no limit.
(negative infinity, 0)U(0, positive infinity)
 
According to one of my fellow Wyzants, Michael F. of Wilton, Connecticut, "the parentheses ( and ) indicate no endpoints. The symbol ∪ means set union."

Comments

I know the answer is e1.  Your answer does not make sense to me? I was looking more so on how to get the answer. I'm pretty sure you're giving me the wrong answer.
Are you asking for the limit of the results of this equation?
Or are you asking for the limit of the value of x?