Find the limit (if it exists). Note that represents the greatest integer function.

Question

Find the limit (if it exists). Note that  represents the greatest integer function.

Quizplus · Accepted Answer

We can approach this problem by using sandwich theorem. We know that $0 \leq \sin(\frac{1}{n}) \leq 1$ for all $n$, and thus $-5 \leq -5\sin(\frac{1}{n}) \leq 5$. Also, as $n ightarrow \infty$, $11ec1854_fdca_aba4_acfc_65bc54da40f7_TBX8702_11(\frac{n}{\pi})$ will approach a multiple of $\frac{1}{\pi}$ (since for large $n$, $\frac{n}{\pi}$ will be very close to some integer and $11ec1854_fdca_aba4_acfc_65bc54da40f7_TBX8702_11$ rounds down), namely $\lfloor\frac{n}{\pi}floor \cdot \frac{1}{\pi}$. Thus, we see that 
$$-5\leq -5\sin\left(\frac{1}{n}ight) \leq \lfloor\frac{n}{\pi}floor \cdot \frac{1}{\pi}\leq \frac{n}{\pi}\cdot \frac{1}{\pi} \leq 5\sin\left(\frac{1}{n}ight) \leq 5$$
Taking the limit as $n	o\infty$, we see that the left and right sides of the inequality approach $-5$ and $5$ respectively, while the middle approaches the limit which is the same as the limit of $\lfloor\frac{n}{\pi}floor$, which is $\lfloor\frac{1}{\pi}floor=0$. Hence, the limit is $\boxed{\mathrm{(D) } 16}.$

Find the Limit (If It Exists)