Find the limit (if it exists). Note that represents the greatest integer function.
A)17
B)15
C)0
D)16
E)Limit does not exist.

Question

Find the limit (if it exists). Note that  represents the greatest integer function. 
A)17 
B)15 
C)0 
D)16 
E)Limit does not exist.

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)