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).