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The given problem is a limit:

\[
\lim_{{x \to 0}} \frac{3 - 3 \cos^2(x)}{6 \tan^2(x)}
\]

### Step-by-step solution:

1. **Rewrite the trigonometric functions in simpler terms:**
   The expression involves \( \cos^2(x) \) and \( \tan^2(x) \). Using the identity \( \tan(x) = \frac{\sin(x)}{\cos(x)} \), we can rewrite \( \tan^2(x) \) as:
   
   \[
   \tan^2(x) = \frac{\sin^2(x)}{\cos^2(x)}
   \]

   So the limit becomes:

   \[
   \lim_{{x \to 0}} \frac{3 - 3 \cos^2(x)}{6 \cdot \frac{\sin^2(x)}{\cos^2(x)}}
   \]

2. **Simplify the denominator:**
   The denominator becomes:

   \[
   6 \cdot \frac{\sin^2(x)}{\cos^2(x)} = \frac{6 \sin^2(x)}{\cos^2(x)}
   \]

   Hence, the limit expression is now:

   \[
   \lim_{{x \to 0}} \frac{3(1 - \cos^2(x))}{\frac{6 \sin^2(x)}{\cos^2(x)}}
   \]

3. **Use the identity \( 1 - \cos^2(x) = \sin^2(x) \):**
   Applying this trigonometric identity, the numerator becomes:

   \[
   3(1 - \cos^2(x)) = 3 \sin^2(x)
   \]

   The limit expression is now:

   \[
   \lim_{{x \to 0}} \frac{3 \sin^2(x)}{\frac{6 \sin^2(x)}{\cos^2(x)}}
   \]

4. **Simplify the fraction:**
   Cancel out \( \sin^2(x) \) from both the numerator and the denominator:

   \[
   \lim_{{x \to 0}} \frac{3}{\frac{6}{\cos^2(x)}} = \lim_{{x \to 0}} \frac{3 \cos^2(x)}{6}
   \]

   Simplifying this further:

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

5. **Evaluate the limit:**
   As \( x \to 0 \), \( \cos(x) \to 1 \). Therefore, the limit becomes:

   \[
   \frac{1^2}{2} = \frac{1}{2}
   \]

### Final Answer:
\[
\lim_{{x \to 0}} \frac{3 - 3 \cos^2(x)}{6 \tan^2(x)} = \frac{1}{2}
\]

Would you like more details on any step, or have any questions?

### Follow-up Questions:
1. What are other common trigonometric identities used in limits?
2. How does L'Hopital's Rule help in solving indeterminate limits like \( \frac{0}{0} \)?
3. Can this limit be solved using Taylor series expansion?
4. What other forms of limits result in \( \frac{0}{0} \) indeterminate forms?
5. How do small-angle approximations simplify limits involving trigonometric functions?

### Tip:
For limits involving trigonometric functions as \( x \to 0 \), it's often helpful to apply small-angle approximations, such as \( \sin(x) \approx x \) and \( \cos(x) \approx 1 \).

lim (x -> 0) (3 - 3cos^2(x)) / (6tan^2(x))

Learn how to evaluate the limit lim (x -> 0) (3 - 3cos^2(x)) / (6tan^2(x)) with detailed steps, covering key trigonometric identities and simplifications. Suitable for students in Grades 11-12 or Introductory Calculus courses.

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