lim x -> 0 (3 - 3cos^2 x)/(x ^ 2) adalah...

We are tasked with finding the limit:

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

### Step 1: Factor out the constant
We can factor out the constant \(3\) from the numerator:

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

### Step 2: Use a trigonometric identity
Recall that \(1 - \cos^2 x = \sin^2 x\) (from the Pythagorean identity). Substituting this into the expression:

\[
\lim_{x \to 0} \frac{3\sin^2 x}{x^2}
\]

### Step 3: Simplify the limit
We know that \(\lim_{x \to 0} \frac{\sin x}{x} = 1\), so we can rewrite the expression:

\[
\lim_{x \to 0} 3 \left( \frac{\sin x}{x} \right)^2 = 3 \times 1^2 = 3
\]

Thus, the value of the limit is:

\[
\boxed{3}
\]

Would you like further clarification on any step?

Here are five related questions you can explore:
1. What is the limit of \(\lim_{x \to 0} \frac{1 - \cos x}{x^2}\)?
2. How can we derive \(\lim_{x \to 0} \frac{\sin x}{x}\) using L'Hopital's Rule?
3. What are some common trigonometric limits and how are they derived?
4. How does L'Hopital's Rule apply to limits involving indeterminate forms?
5. What is the Maclaurin series for \(\sin x\) and \(\cos x\), and how can it be used in limits?

**Tip:** For small angles, trigonometric functions like \(\sin x\) and \(\cos x\) can often be approximated using Taylor or Maclaurin series expansions.

lim x -> 0 (3 - 3cos^2 x)/(x^2) adalah...

Explore the detailed solution to finding the limit of (3 - 3cos^2 x)/x^2 as x approaches 0. This problem utilizes trigonometric identities and the limit of sin(x)/x for a step-by-step approach, suitable for students in Grades 11-12.

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