Finding the Limit of a Root Test: Calculating the Limit of a Challenging Term

[MissP.25_5]

Hello.
How do I find the limit of this term?

$$\lim_{{n}\to{\infty}}|\left(\frac{n}{n+1}\right)^{\!{n^2}}|^\frac{1}{n}$$

This is the working but I don't understand how to get the third line.

r = lim(n→∞) |[n/(n+1)]^(n^2)|^(1/n)
..= lim(n→∞) [n/(n+1)]^n
..= lim(n→∞) 1 / [(n+1)/n]^n
..= lim(n→∞) 1 / (1 + 1/n)^n
..= 1/e, by the limit definition of e.

 

[Erland]

$\frac 1{\frac{n+1}n}=\frac n{n+1}$ and $(\frac 1a)^n = \frac 1{a^n}$, right?

 

[MissP.25_5]

$\frac 1{\frac{n+1}n}=\frac n{n+1}$ and $(\frac 1a)^n = \frac 1a^n$, right?

Yes, but as you can see, the third line has its denominator to the power of n. That's what I don't understand. If we divide all terms with n, then what happens to the n outside the bracket?

 

[Erland]

Sorry, there was a typo in my reply which I corrected immediatly, but you were so quick and got the typo in your reply...

 

[MissP.25_5]

Sorry, there was a typo in my reply which I corrected immediatly, but you were so quick and got the typo in your reply...

Yeah, I was quick, haha. As soon as I submitted my post, I saw you have corrected your error. Thank you!