real analysis - Find the limit of this sequence $lim_{nto infty}frac{n}{1 + frac{1}{n}} - n$

Find the limit of this sequence $$\lim_{n\to \infty}\frac{n}{1 + \frac{1}{n}} - n$$

First I tried dividing everything by $n$ but that would leave me with $$\lim_{n\to \infty}\frac{1}{\frac{1}{n} + \frac{1}{n^2}} - 1$$

and as $n\to \infty$ i'd be left with $\frac{1}{0} - 1$. Would I be correct in saying that the limit is -1 or does the $\frac{1}{0}$ mess that up?

Answer

The limit is $-1$ but the $\frac{1}{0}$ does mess it up.

Just add the fractions:

you get $$\frac{n^2}{n+1} - n = \frac{n^2 - n^2 - n}{n} = \frac{-n}{n+1}$$

Finding the limit should now be easy