Amanita, let's focus on the problem inside the green rectangle:

$\lim_{n \to \infty} \left(1 + \frac{1}{7^n}\right)$

Here’s the step-by-step plan to solve this problem:

Step 1: Analyze the expression inside the limit.
We have $1 + \frac{1}{7^n}$. As $n$ gets larger and larger, $7^n$ becomes a very big number.

Step 2: Understand the behavior of $\frac{1}{7^n}$ as $n \to \infty$.
When the denominator grows very large, the whole fraction $\frac{1}{7^n}$ gets closer and closer to 0.

Step 3: Substitute the behavior into the original expression.
As $n$ goes to infinity, $\frac{1}{7^n}$ goes to 0, so the expression inside the limit becomes $1 + 0$.

Step 4: Write the final answer.
The limit is simply 1.

Now, let's confirm this by calculating the limit:

$\lim_{n \to \infty} \left(1 + \frac{1}{7^n}\right)$

Amanita, after following the steps and calculating the limit, we find:

$\lim_{n \to \infty} \left(1 + \frac{1}{7^n}\right) = 1$

As $n$ becomes very large, $\frac{1}{7^n}$ gets closer and closer to 0, so the whole expression approaches 1. The answer is 1.