Question

Evaluate: $\underset{n\to \infty }{lim}(n-\sqrt{n^2+n})$.

Answer 1

$\underset{n\to \infty }{lim}\frac{(n-\sqrt{n^2+n})(n+\sqrt{n^2+n})}{(n+\sqrt{n^2+n})}$

$=\underset{n\to \infty }{lim}\frac{n^2-n^2-n}{n+\sqrt{n^2+n}}$

$=\underset{n\to \infty }{lim}\frac{-1}{1+\sqrt{1+1/n}}$

$=\frac{-1}{1+1}=\frac{-1}{2}$.