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 $\cos(x) \approx 1$.