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We are asked to calculate the limit:

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

### Step-by-step solution:

1. **Recognize the indeterminate form**:  
   As \( x \to 0 \), the numerator approaches 0, and the denominator also approaches 0 since \( 1 - \cos(0) = 0 \). This is an indeterminate form of type \( \frac{0}{0} \), so we can apply **L'Hôpital's Rule**, which is used to evaluate limits of this indeterminate form by differentiating the numerator and the denominator.

2. **Differentiate the numerator**:
   \[
   \text{Numerator: } -3x^2 \implies \frac{d}{dx}(-3x^2) = -6x
   \]

3. **Differentiate the denominator**:
   \[
   \text{Denominator: } 1 - \cos(x) \implies \frac{d}{dx}(1 - \cos(x)) = \sin(x)
   \]

4. **Apply L'Hôpital's Rule**:  
   Now, we can rewrite the limit as:
   \[
   \lim_{x \to 0} \frac{-6x}{\sin(x)}
   \]

5. **Simplify the limit**:  
   We know the standard limit \( \lim_{x \to 0} \frac{x}{\sin(x)} = 1 \). Using this, we have:
   \[
   \lim_{x \to 0} \frac{-6x}{\sin(x)} = -6 \times \lim_{x \to 0} \frac{x}{\sin(x)} = -6 \times 1 = -6
   \]

Thus, the limit is:

\[
\boxed{-6}
\]

---

Would you like more details or have any other questions? Here are some related questions to expand your understanding:

1. What is L'Hôpital's Rule, and when can it be applied?
2. How can we evaluate limits involving trigonometric functions without L'Hôpital's Rule?
3. What are common trigonometric limits like \( \lim_{x \to 0} \frac{\sin x}{x} \)?
4. Can L'Hôpital's Rule be applied multiple times if the indeterminate form persists?
5. How do we handle indeterminate forms of other types, such as \( \frac{\infty}{\infty} \)?

**Tip**: Always check if you can simplify the limit expression before directly applying L'Hôpital’s Rule, as sometimes algebraic simplifications can make the process easier.

Calculate the limit: lim (x -> 0) of (-3x^2)/(1 - cos x)

Learn how to calculate the limit of (-3x^2) / (1 - cos x) as x approaches 0 by applying L'Hopital's Rule. This solution uses differentiation and standard trigonometric limits to simplify the problem. Suitable for Grade 12 or early college students.

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