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Let's solve the given problems step by step.

### Problem 1:
\[ \lim_{x \to 0} \frac{\sin x + \sin 2x + \ldots + \sin nx}{x} = 15 \]

This expression can be simplified by recognizing that near \( x = 0 \), \( \sin(kx) \approx kx \). Therefore, we approximate each term:

\[ \sin(kx) \approx kx \]

Summing the series:

\[ \sin x + \sin 2x + \ldots + \sin nx \approx x + 2x + \ldots + nx = x(1 + 2 + \ldots + n) \]

The sum of the first \( n \) natural numbers is given by:

\[ 1 + 2 + \ldots + n = \frac{n(n+1)}{2} \]

Thus,

\[ \sin x + \sin 2x + \ldots + \sin nx \approx x \cdot \frac{n(n+1)}{2} \]

Substituting this back into the limit expression, we get:

\[ \lim_{x \to 0} \frac{x \cdot \frac{n(n+1)}{2}}{x} = \frac{n(n+1)}{2} \]

We know from the problem statement that this equals 15:

\[ \frac{n(n+1)}{2} = 15 \]

Multiplying both sides by 2:

\[ n(n+1) = 30 \]

To find \( n \), solve the quadratic equation:

\[ n^2 + n - 30 = 0 \]

Using the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 1 \), and \( c = -30 \):

\[ n = \frac{-1 \pm \sqrt{1 + 120}}{2} = \frac{-1 \pm \sqrt{121}}{2} = \frac{-1 \pm 11}{2} \]

This gives us two solutions:

\[ n = \frac{-1 + 11}{2} = 5 \]
\[ n = \frac{-1 - 11}{2} = -6 \] (not applicable since \( n \) must be positive)

Therefore, \( n = 5 \).

### Problem 2:
\[ x_{n+1} = \frac{1}{2} \left( x_n + \frac{a}{x_n} \right), \quad x_1 > 0 \]
Determine \( \lim_{n \to \infty} x_n \)

Assume \( \lim_{n \to \infty} x_n = L \). Then:

\[ L = \frac{1}{2} \left( L + \frac{a}{L} \right) \]

Multiplying both sides by 2:

\[ 2L = L + \frac{a}{L} \]

Subtract \( L \) from both sides:

\[ L = \frac{a}{L} \]

Multiplying both sides by \( L \):

\[ L^2 = a \]

Therefore:

\[ L = \sqrt{a} \]

Thus,

\[ \lim_{n \to \infty} x_n = \sqrt{a} \]

If you have any questions or need further details, feel free to ask.

### Questions to consider:
1. How can the approximation \( \sin(kx) \approx kx \) be justified more rigorously?
2. What are the steps involved in deriving the sum of the first \( n \) natural numbers?
3. How can we solve quadratic equations using the quadratic formula?
4. What other methods can be used to solve the quadratic equation \( n^2 + n - 30 = 0 \)?
5. How does the sequence \( x_{n+1} = \frac{1}{2} \left( x_n + \frac{a}{x_n} \right) \) relate to the arithmetic-geometric mean?
6. What are the properties of sequences that converge to a limit?
7. How can limits of sequences be determined using recursive relations?
8. How does the convergence of the sequence depend on the initial value \( x_1 \)?

### Tip:
When solving limits involving trigonometric functions, using series expansion or small-angle approximations can greatly simplify the problem.

Explore advanced mathematical problem-solving techniques involving limits, trigonometric functions, series approximations, quadratic equations, and recursive sequences. Detailed solutions to specific problems included. Suitable for advanced high school students and beyond.

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