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Let's go through each question in detail.

### Question 1

We are given the function:
\[
f(x) = \frac{e^{-x}}{x} + 5
\]
and the sequence:
\[
a_n = \frac{e^{-n}}{n} + 5
\]

We need to determine the behavior of \( f(x) \) as \( x \to \infty \) and make conclusions about the convergence of the sequence \( a_n \).

1. **Limit of \( f(x) \) as \( x \to \infty \):**
   - As \( x \to \infty \), the term \( e^{-x} \) (which is \( \frac{1}{e^x} \)) approaches 0 very quickly. Therefore, \( \frac{e^{-x}}{x} \to 0 \) as \( x \to \infty \).
   - This makes \( f(x) \to 5 \) as \( x \to \infty \).

2. **Behavior of the sequence \( a_n \):**
   - Since \( a_n = \frac{e^{-n}}{n} + 5 \), as \( n \to \infty \), \( \frac{e^{-n}}{n} \) will also approach 0 (due to the exponential decay of \( e^{-n} \)).
   - Thus, \( a_n \to 5 \) as \( n \to \infty \), indicating that the sequence converges to 5.

Therefore, the correct answer is:
- **The limit of \( f(x) \) as \( x \to \infty \) is \( L = 5 \), and the sequence \( a_n \) will converge to the same limit.**

### Question 2

The sequence is defined recursively as:
\[
a_{n+1} = \frac{3}{4}a_n - 1, \quad a_0 = -1
\]

Let's examine the behavior of this sequence.

1. **Recursive Analysis:**
   - We can calculate a few terms to see the pattern:
     - \( a_0 = -1 \)
     - \( a_1 = \frac{3}{4}(-1) - 1 = -\frac{3}{4} - 1 = -\frac{7}{4} \)
     - \( a_2 = \frac{3}{4} \left(-\frac{7}{4}\right) - 1 = -\frac{21}{16} - 1 = -\frac{37}{16} \)
   - The sequence appears to be decreasing.

2. **Long-term behavior:**
   - Since \( \left| \frac{3}{4} \right| < 1 \), this sequence will tend towards a stable value as \( n \to \infty \).
   - A sequence defined by \( a_{n+1} = r a_n + c \) (where \( |r| < 1 \)) tends to the fixed point \( L = \frac{c}{1 - r} \).
   - Here, \( r = \frac{3}{4} \) and \( c = -1 \), so the fixed point is:
     \[
     L = \frac{-1}{1 - \frac{3}{4}} = \frac{-1}{\frac{1}{4}} = -4
     \]

Thus, the sequence converges to \(-4\) and is **decreasing and bounded below**.

The correct answer is:
- **Decreasing and bounded below.**

---

Would you like additional details or have any questions?

### Related Questions

1. How would the solution change if \( f(x) = \frac{e^{-x^2}}{x} + 5 \) instead?
2. What if \( a_n = \frac{e^{-n^2}}{n} + 5 \); would it converge to the same limit?
3. How can you determine if a sequence is bounded or unbounded?
4. For recursive sequences, how does the initial term \( a_0 \) affect long-term behavior?
5. Can recursive sequences have multiple fixed points?

### Tip
In recursive sequences, a constant ratio with \( |r| < 1 \) typically leads to convergence towards a steady-state (fixed point).

Calculate the following limit of the function f as x approaches infinity. What can you conclude about the convergence of the sequence a_n?

Explore how to calculate the limit of a function as x approaches infinity and determine the convergence of the sequence a_n. This problem covers fundamental calculus concepts on limits and sequences, providing insight into convergence behavior.

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