Find the linit of the following sequence or determine that the sequience diverges $$ \left\{\frac{2^{n+1}+3}{2^{n}}\right\} $$ Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The limit of the sequence is B. The (Type an exact answer.) Bequence diverges.

Question

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We are given the sequence

$$
a_n = \frac{2^{n+1}+3}{2^n}.
$$

**Step 1. Simplify the expression**

Notice that

$$
2^{n+1} = 2 \cdot 2^n.
$$

Thus, we can rewrite the sequence as

$$
a_n = \frac{2 \cdot 2^n + 3}{2^n} = \frac{2 \cdot 2^n}{2^n} + \frac{3}{2^n} = 2 + \frac{3}{2^n}.
$$

**Step 2. Find the limit as $$ n \to \infty $$**

The limit we need to compute is

$$
\lim_{n \to \infty} a_n = \lim_{n \to \infty} \left(2 + \frac{3}{2^n}\right).
$$

Since $$ \lim_{n \to \infty} \frac{3}{2^n} = 0 $$ (because $$ 2^n $$ grows without bound), we have

$$
\lim_{n \to \infty} a_n = 2 + 0 = 2.
$$

**Final Answer**

A. The limit of the sequence is 2.
The limit of the sequence is 2.

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