Question

Evaluate ${lim}_{n\to \infty }\sin \left(\pi \sqrt{n^2+0.5n+1}\right)$, where $nϵN$.

Answer 1

$⟶\underset{n\to \infty }{lim}\sin (\pi \sqrt{n^2+0.5n+1)}$

$=\underset{n\to \infty }{lim}\sin \left(\pi \sqrt{n^2(1+\frac{0.5n}{n^2}+\frac{1}{n^2})}\right)$

$=\underset{n\to \infty }{lim}\sin (\pi \sqrt{n^2(1+0+0))}$

$=\underset{n\to \infty }{lim}\sin \pi n$

$=0$