What is the limit of this question?

Question

Lim x-->∞  (1+¹/_x)^x

Answer 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

lim_x→∞ y = lim_x→∞ ln(1+1/x) / (1/x)

which is of the form 0/0, so by l'Hospital's Rule,

lim_x→∞ 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!

 

Answer 2

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 :)

Answer 3

limit of (1 + 1/x)^x as x ---> ∞ is e.  e ≈ 2.71828........

Answer 4

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."