The number e as a limit

Homework Statement

I have lim of n > infinity (1+1/n)^n

The Attempt at a Solution

I know that I must use l'hospital rule and setting ln y = n ln (1+1/n)

And after lim n ln (1+1/n) as n approaches infinity.

After what do I do ?

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LCKurtz
Homework Helper
Gold Member

Homework Statement

I have lim of n > infinity (1+1/n)^n

The Attempt at a Solution

I know that I must use l'hospital rule and setting ln y = n ln (1+1/n)

And after lim n ln (1+1/n) as n approaches infinity.

After what do I do ?
Try writing it as$$\frac{\ln(1 + \frac 1 n)}{\frac 1 n}$$before using L'Hospital's rule and taking the limit.

Try writing it as$$\frac{\ln(1 + \frac 1 n)}{\frac 1 n}$$before using L'Hospital's rule and taking the limit.
OK but I'll have to take the natural logarithm right ?

LCKurtz
Homework Helper
Gold Member
OK but I'll have to take the natural logarithm right ?
Didn't you already take the logarithm to get that expression?

Student100
Gold Member

Homework Statement

I have lim of n > infinity (1+1/n)^n

The Attempt at a Solution

I know that I must use l'hospital rule and setting ln y = n ln (1+1/n)

And after lim n ln (1+1/n) as n approaches infinity.

After what do I do ?
You've either applied LH incorrectly, or I'm misreading you. How did you apply LH (how did you set it up) when you're at the step: $$exp[\lim_{n\to\infty}(n\log(1+\frac{1}{n})]$$

Never mind, didn't see the new post update. :) Ignore me LC's already said the same thing.

You've either applied LH incorrectly, or I'm misreading you. How did you apply LH (how did you set it up) when you're at the step: $$exp[\lim_{n\to\infty}(n\log(1+\frac{1}{n})]$$

Never mind, didn't see the new post update. :) Ignore me LC's already said the same thing.
Wait, what I just wrote was correct or not ? Am I in the right direction ?

LCKurtz
Homework Helper
Gold Member
Wait, what I just wrote was correct or not ? Am I in the right direction ?

Mark44
Mentor

Homework Statement

I have lim of n > infinity (1+1/n)^n

The Attempt at a Solution

I know that I must use l'hospital rule and setting ln y = n ln (1+1/n)

And after lim n ln (1+1/n) as n approaches infinity.

After what do I do ?
For this to work out correctly, you need to keep track of your equations. Each line you write should be an equation.
Let $y = (1 + 1/n)^n$
$\ln y = n \ln(1 + 1/n) = \frac{\ln(1 + 1/n)}{1/n}$
Now take limits of both sides, after which you can apply L'Hopital's Rule.

Again, take care to work with equations at each step.

For this to work out correctly, you need to keep track of your equations. Each line you write should be an equation.
Let $y = (1 + 1/n)^n$
$\ln y = n \ln(1 + 1/n) = \frac{\ln(1 + 1/n)}{1/n}$
Now take limits of both sides, after which you can apply L'Hopital's Rule.

Again, take care to work with equations at each step.
I'm sorry if it takes time to respond, I have like several question to answer at the same and I must give this tomorrow lol Don't worry, I keep track of what I write on paper. I'll respond here if something I'm having a problem with.

Ok thanks it worked

I'm sorry if it takes time to respond, I have like several question to answer at the same and I must give this tomorrow lol Don't worry, I keep track of what I write on paper. I'll respond here if something I'm having a problem with.
Yes, but it is extremely rude and time consuming for the people on this site. How are we to know the work you present means.