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# What is the limit of this question?

Lim x-->∞  (1+1/x)x

### 4 Answers by Expert Tutors

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. | Summer Online Finance Specialist: WACC NPV DCF TVM YTMSummer Online Finance Specialist: WACC ...
4.9 4.9 (233 lesson ratings) (233)
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........

Can you please explain how you derive this answer? I know you are correct.

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."