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
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