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The given problem defines a sequence \( x_n \) as follows:

- \( x_0 = 0 \)
- \( x_{n+1} = x_n - \frac{\tan x_n - 1}{\sec^2 x_n} \)

You are asked to find \( \lim_{n \to \infty} x_n \), if it exists, and to relate this to Newton's method.

### Step 1: Understanding the Recurrence Relation

The recurrence relation can be written as:
\[
x_{n+1} = x_n - \frac{\tan(x_n) - 1}{\sec^2(x_n)}
\]
Since \( \sec^2(x) = 1 + \tan^2(x) \), we can rewrite the equation:
\[
x_{n+1} = x_n - \frac{\tan(x_n) - 1}{1 + \tan^2(x_n)}
\]

### Step 2: Relating to Newton's Method

Newton's method is used to approximate the roots of a function \( f(x) = 0 \). The general formula for Newton's method is:
\[
x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}
\]

Comparing this with the recurrence relation provided, we can infer that this is Newton's method applied to the function:
\[
f(x) = \tan(x) - 1
\]
Thus, the iterative formula for Newton's method becomes:
\[
x_{n+1} = x_n - \frac{\tan(x_n) - 1}{\sec^2(x_n)}
\]
This confirms that the given recurrence is Newton's method applied to \( f(x) = \tan(x) - 1 \).

### Step 3: Finding the Limit

Newton's method typically converges to a root of the equation \( f(x) = 0 \). The equation \( \tan(x) = 1 \) has solutions:
\[
x = \frac{\pi}{4} + n\pi \quad \text{for integer } n
\]
Since \( x_0 = 0 \), the sequence likely converges to the smallest positive solution of \( \tan(x) = 1 \), which is:
\[
x = \frac{\pi}{4}
\]

Thus, the limit is:
\[
\lim_{n \to \infty} x_n = \frac{\pi}{4}
\]

### Conclusion:

- The sequence \( x_n \) converges to \( \frac{\pi}{4} \).
- This recurrence relation is Newton's method applied to the function \( f(x) = \tan(x) - 1 \), and it converges to the root \( x = \frac{\pi}{4} \), where \( \tan(x) = 1 \).

---

Would you like more detailed steps for Newton's method or further clarification? Here are 5 related questions to explore:

1. How does Newton's method work in general for finding roots?
2. Why does Newton's method fail in certain cases?
3. What are the conditions for Newton's method to converge?
4. Can Newton's method be applied to complex functions as well?
5. How do we choose a good initial guess \( x_0 \) in Newton's method?

**Tip**: Newton's method converges faster when the derivative \( f'(x) \) is not too small near the root.

Define x_0 = 0 and x_{n+1} = x_n - [(tan(x_n) - 1) / sec^2(x_n)]. What is lim_{n→∞} x_n in this example? Relate this to Newton's method.

Explore how Newton's method is applied to the function tan(x) - 1, with a step-by-step solution showing how the recurrence relation converges to π/4. This detailed example covers key concepts like limits, trigonometric identities, and iterative methods for root-finding.

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