I can help you solve these limit problems. Let's extract and go through each question step by step.

The questions are:

1. $\lim_{x \to \frac{\pi}{3}} \frac{4 \tan x - 2 \sin x}{4 \cos x}$
2. $\lim_{x \to -\frac{\pi}{4}} \frac{\sin 3x}{\sin 3x + \cos x}$
3. $\lim_{x \to 0} \frac{\sin^2 2x}{2x^2}$
4. $\lim_{x \to 0} \frac{\sin^2 3x}{\tan^2 \frac{1}{2}x}$
5. $\lim_{x \to 0} \frac{4 \sin^2 2x}{x \tan 2x (\sin x - \cos x)}$

I will now solve each one. Let's start with the first limit.The solution to the first limit is:

$\lim_{x \to \frac{\pi}{3}} \frac{4 \tan x - 2 \sin x}{4 \cos x} = \frac{3\sqrt{3}}{2}$

Now, let's move on to the second problem.The second limit does not exist because it approaches infinity as $x \to -\frac{\pi}{4}$:

$\lim_{x \to -\frac{\pi}{4}} \frac{\sin 3x}{\sin 3x + \cos x} = \infty$

Let's proceed to the third problem.The solution to the third limit is:

$\lim_{x \to 0} \frac{\sin^2 2x}{2x^2} = 2$

Now, let's solve the fourth problem.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like me to attempt a different approach, or do you have any other questions? Let me know how you'd like to proceed.

In the meantime, here are some related questions you might find useful:

1. How do limits behave for trigonometric functions at specific points?
2. What are common techniques for solving limits involving sine and cosine?
3. How does the squeeze theorem help in solving limits of trigonometric functions?
4. Can indeterminate forms like $\frac{0}{0}$ be resolved using L'Hopital's Rule?
5. What role do small-angle approximations play in limit calculations for sine and tangent?

Tip: When solving limits involving trigonometric functions, it’s often useful to apply standard trigonometric identities or small-angle approximations to simplify expressions.