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The given limit problem is:

\[
\lim_{{x \to 0}} \frac{1 - \cos(3x)}{\cos^2(5x) - 1}
\]

### Step 1: Analyze the denominator
We know that \(\cos^2(5x) - 1\) is a standard trigonometric identity. Using the identity \(\cos^2(\theta) - 1 = -\sin^2(\theta)\), we rewrite the denominator:

\[
\cos^2(5x) - 1 = -\sin^2(5x)
\]

Thus, the limit becomes:

\[
\lim_{{x \to 0}} \frac{1 - \cos(3x)}{-\sin^2(5x)}
\]

### Step 2: Use small-angle approximations
For small \(x\), we use the standard approximations:
- \(\cos(x) \approx 1 - \frac{x^2}{2}\)
- \(\sin(x) \approx x\)

Using this for \(\cos(3x)\) and \(\sin(5x)\):

- \(1 - \cos(3x) \approx 1 - \left(1 - \frac{(3x)^2}{2}\right) = \frac{9x^2}{2}\)
- \(\sin^2(5x) \approx (5x)^2 = 25x^2\)

Substituting these into the limit expression:

\[
\lim_{{x \to 0}} \frac{\frac{9x^2}{2}}{-25x^2}
\]

### Step 3: Simplify the expression
The \(x^2\) terms cancel out, leaving:

\[
\lim_{{x \to 0}} \frac{9}{-50} = -\frac{9}{50}
\]

Thus, the value of the limit is:

\[
\boxed{-\frac{9}{50}}
\]

Let me know if you need more details or have any questions!

Here are 5 related questions to expand on this concept:
1. How do you apply L'Hopital's Rule to limits involving trigonometric functions?
2. What is the Taylor series expansion of \(\cos(x)\) around \(x = 0\)?
3. Why does the small-angle approximation work for limits as \(x \to 0\)?
4. How does the limit change if the trigonometric functions had different coefficients for \(x\)?
5. What are some other useful trigonometric identities for solving limits?

**Tip:** When dealing with limits involving trigonometric functions as \(x \to 0\), small-angle approximations are powerful tools to simplify the calculations.

Solve the limit: lim (x→0) [(1 - cos(3x)) / (cos^2(5x) - 1)]

A step-by-step guide to solving the limit problem involving trigonometric functions: lim (x→0) [(1 - cos(3x)) / (cos^2(5x) - 1)]. This solution uses trigonometric identities and small-angle approximations for simplification. Perfect for advanced high school or early college-level students.

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