https://r2.erweima.ai/mathbot/image/1726186776140txrxofiv.jpg

Let's break down the expression in the image into manageable parts:

\[
\left( \lim_{{x \to 12}} \frac{{x^2 - 25}}{{x + 5}} \right)^9 - \left( \sum_{{k=0}}^{\infty} \left( \frac{6}{7} \right)^k \right)^8 + \left( \frac{{-125.4387e^{\pi i}}}{{\lim_{{x \to 25}} (\cos^2 x + \sin^2 x)}} \right)
\]

### Step-by-step solution:

1. **First limit**: \(\lim_{{x \to 12}} \frac{{x^2 - 25}}{{x + 5}}\)

   Factorize the numerator:
   \[
   x^2 - 25 = (x - 5)(x + 5)
   \]
   Therefore, the limit becomes:
   \[
   \lim_{{x \to 12}} \frac{{(x - 5)(x + 5)}}{{x + 5}} = \lim_{{x \to 12}} (x - 5)
   \]
   Substitute \(x = 12\):
   \[
   12 - 5 = 7
   \]
   So, the first part becomes \(7^9\).

2. **Geometric series sum**: \(\sum_{{k=0}}^{\infty} \left( \frac{6}{7} \right)^k\)

   This is a geometric series with the first term \(a = 1\) and the common ratio \(r = \frac{6}{7}\). The sum of an infinite geometric series is given by:
   \[
   S = \frac{a}{1 - r} = \frac{1}{1 - \frac{6}{7}} = \frac{1}{\frac{1}{7}} = 7
   \]
   So, the second part becomes \(7^8\).

3. **Third limit**: \(\lim_{{x \to 25}} (\cos^2 x + \sin^2 x)\)

   By the Pythagorean identity, we know that:
   \[
   \cos^2 x + \sin^2 x = 1 \quad \text{for any} \ x
   \]
   Thus, the limit is simply 1.

4. **Euler's identity**: \(e^{\pi i}\)

   Euler's identity states that:
   \[
   e^{\pi i} = -1
   \]
   Therefore, the term \(-125.4387e^{\pi i}\) simplifies to:
   \[
   -125.4387 \times (-1) = 125.4387
   \]

### Putting it all together:

- The first part is \(7^9 = 40353607\),
- The second part is \(7^8 = 5764801\),
- The third part simplifies to \( \frac{125.4387}{1} = 125.4387\).

Thus, the entire expression simplifies to:
\[
40353607 - 5764801 + 125.4387 = 34588831.4387
\]

The final result is approximately:
\[
34,588,831.44
\]

---

Would you like more details or clarifications on any part? Here are some related questions you might find interesting:

1. What is the formula for the sum of a geometric series?
2. How do you simplify limits that involve factorization?
3. Can you explain why \(\cos^2 x + \sin^2 x = 1\)?
4. How is Euler's identity derived from complex numbers?
5. What other identities are related to Euler's formula?

**Tip:** When simplifying expressions involving limits and series, always look for identities or factorization that can make the problem easier.

Evaluate the limit expression involving a quadratic fraction, an infinite sum, and Euler's identity.

This solution evaluates a complex mathematical expression involving limits, geometric series, and Euler's identity. It covers key concepts such as the Pythagorean identity and infinite series summation, suitable for advanced calculus students.

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