# Thread: Limit at infinity involving e

1. ## Limit at infinity involving e

I'm having some troubles with this one:

lim x -> +infinity of (x + 1)^x / (x^x)

The answer in the back of the book says the limit is e. So I figure that somehow I need to get the denominator to be 1 and the numerator to be in the form (1 + 1/x)^x. But I can't seem to get that. Is there some other way? Or maybe I'm missing a step. I've tried dividing top and bottom by x and x^x but this x exponent is messing me up algebraically.

2. Originally Posted by nautica17
I'm having some troubles with this one:

lim x -> +infinity of (x + 1)^x / (x^x)

The answer in the back of the book says the limit is e. So I figure that somehow I need to get the denominator to be 1 and the numerator to be in the form (1 + 1/x)^x. But I can't seem to get that. Is there some other way? Or maybe I'm missing a step. I've tried dividing top and bottom by x and x^x but this x exponent is messing me up algebraically.

$\displaystyle \frac{(x+1)^x}{x^x} = \left( \frac{x+1}{x} \right) ^x = \left( 1+ \frac{1}{x} \right) ^x$
$\displaystyle \frac{(x+1)^x}{x^x} = \left( \frac{x+1}{x} \right) ^x = \left( 1+ \frac{1}{x} \right) ^x$