Limit of a certain function of n as n goes to infinity

[ShizukaSm]

Homework Statement

$\lim_{n->\infty} 3(\frac{n}{n+1})^n$

The Attempt at a Solution

Ok, I know that the answer is 3/e, because this limit was solved a year ago when I took calculus 1 by my teacher, and I foolishly copied only the answer, thinking I would never forget and have to go back.

I can't for the life of me understand how he did that, can someone help me, please?

Thanks in advance!

 

[mfb]

$$\frac{n}{n+1} = 1-\frac{1}{n+1}$$
Shifting the index by 1 (and ignoring the 3), the limit becomes $$\lim_{n \to \infty} \left(1-\frac{1}{n}\right)^n \left(1-\frac{1}{n}\right)^{-1}$$ which is easy to study.

 

[jbunniii]

Another way to look at it:
$$3 \left(\frac{n}{n+1}\right)^n = \frac{3}{\left(\frac{n+1}{n}\right)^n} = \frac{3}{\left(1 + \frac{1}{n}\right)^n}$$
The expression in the denominator on the right hand side should look familiar.

 

[ShizukaSm]

Oh yeah, of course! It's a fundamental limit, Thanks to you both, but I have to say that jbunniii's representation made it perfectly clear.