Understanding the Limit as n Approaches Infinity in Calculus

[Nerpilis]

ok I have this limit question that was done in class but i didn't catch it at the time but they grazed over a step where I'm not sure what the reasoning was.
$$\lim_{n \rightarrow \infty} \left( 1 + \frac{1}{n + 1} \right)^{n} = \lim_{n \rightarrow \infty} \left( 1 + \frac{ \frac{1}{n} }{ 1 + \frac{1}{n} } \right)^{n} = e$$
I see the multiplication of one in the form of 1/n over 1/n and i know that $$\lim_{n \rightarrow \infty} \left( 1 + \frac{1}{n} \right)^{n} = e$$ and i can see the similarity and possible substituions...but what happens to the 'n' exponet since it doesn't substitute nicely?

 

[Galileo]

I see no reason to divide the numerator and the denominator by n. You can simply use a substitution like m=n+1. Or write:
$$\lim_{n \rightarrow \infty} \left( 1 + \frac{1}{n + 1} \right)^{n} = \lim_{n \rightarrow \infty} \left( 1 + \frac{1}{n + 1} \right)^{n+1}\cdot \frac{1}{1+\frac{1}{n+1}}$$

 

[Nerpilis]

I think I'm a little more confused now...I do agree that i don't see what dividing by n did to help.