# Find the limit lim_"n→ oo" cosn^3/(2n)- (3n)/(6n+1)?

## ${\lim}_{\text{n→ oo}} \cos {n}^{3} / \left(2 n\right) - \frac{3 n}{6 n + 1}$

Mar 13, 2018

${\lim}_{n \to \infty} \cos \frac{{n}^{3}}{2 n} - \frac{3 n}{6 n + 1} = - \frac{1}{2}$

#### Explanation:

The limits of both addends are finite, and we can evaluate them separately:

${\lim}_{n \to \infty} \cos \frac{{n}^{3}}{2 n} = 0$

as $\left\mid \cos \left({n}^{3}\right) \right\mid \le 1$, so the numerator is bounded and the denominator tends to $+ \infty$.

${\lim}_{n \to \infty} \frac{3 n}{6 n + 1} = {\lim}_{n \to \infty} \frac{3}{6 + \frac{1}{n}} = \frac{3}{6} = \frac{1}{2}$

Then:

${\lim}_{n \to \infty} \cos \frac{{n}^{3}}{2 n} - \frac{3 n}{6 n + 1} = 0 - \frac{1}{2} = - \frac{1}{2}$