Sequences Quiz

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1. Find the limit, if it exists, of $a_n = \dfrac{3^{n+2} + 3}{3^n + 5}$.

   (a) $\dfrac{1}{9}$
   (b) $\dfrac{1}{3}$
   (c) $3$
   (d) $9$

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2. Find the $8$th term of $\{5n+2\}_{n=1}^\infty$.

   (a) $42$
   (b) $41$
   (c) $47$
   (d) $37$

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3. Find $$\lim_{n \to \infty} \arctan{\sqrt{n+7e^n}}.$$

   (a) $0$
   (b) $\dfrac{\pi}{4}$
   (c) $1$
   (d) $\dfrac{\pi}{2}$

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4. Find $$\lim_{n \to \infty} \dfrac{(n+1)(2n+1)}{n(n-1)}.$$

   (a) $\infty$
   (b) $0$
   (c) $1$
   (d) $2$

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5. Is $a_n = n + \sin{n}$ bounded? What is $\lim_{n \to \infty} a_n$?

   (a) Bounded between $-1$ and $\dfrac{\pi}{4}$
   (b) Bounded between $0$ and $1$
   (c) Unbounded, and $\lim_{n \to \infty} a_n = \infty$
   (d) Unbounded, and $\lim_{n \to \infty} a_n = -\infty$

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