https://file.aiquickdraw.com/mathbot/file/17263788193107o8v85wm.jpg

Let's analyze and solve each of the two problems shown in the image.

### Problem 1:
We are asked to find the limit:

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

#### Solution:
- Recall that for small \( x \), \( \tan x \approx x \), and \( \cos x \approx 1 - \frac{x^2}{2} \).
- Start by simplifying the expression.

\[
\cos^2 x = (1 - \frac{x^2}{2})^2 \approx 1 - x^2
\]

Thus,

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

Now substitute the approximations into the limit expression:

\[
\lim_{{x \to 0}} \frac{x \cdot x}{-2x^2} = \lim_{{x \to 0}} \frac{x^2}{-2x^2} = \frac{1}{-2} = -\frac{1}{2}
\]

Thus, the answer to the first problem is \( \boxed{A. -\frac{1}{2}} \).

---

### Problem 2:
We are asked to find the limit:

\[
\lim_{{x \to 0}} \left( \frac{\cos(4x) \sin(3x)}{5x} \right)
\]

#### Solution:
- Recall that for small \( x \), \( \sin x \approx x \) and \( \cos x \approx 1 \).

For small \( x \):

\[
\cos(4x) \approx 1 \quad \text{and} \quad \sin(3x) \approx 3x
\]

Substituting these approximations into the limit:

\[
\lim_{{x \to 0}} \frac{(1)(3x)}{5x} = \lim_{{x \to 0}} \frac{3x}{5x} = \frac{3}{5}
\]

Thus, the answer to the second problem is \( \boxed{C. \frac{3}{5}} \).

---

Do you have any questions or want further details on any step? Here are some related questions for practice:

1. What is the approximation for \( \sin x \) and \( \cos x \) when \( x \to 0 \)?
2. How do you evaluate limits involving trigonometric functions at small values of \( x \)?
3. How does the small-angle approximation help in solving limits?
4. Can you explain how L'Hopital's Rule could be used for these problems?
5. What are some common limits involving trigonometric functions?

**Tip:** Always consider using small-angle approximations for trigonometric functions when working with limits as \( x \to 0 \).

1. Nilai dari lim (x -> 0) x tan(x) / (2 cos^2(x) - 2) adalah .... 
2. Nilai dari lim (x -> 0) (cos(4x) sin(3x)) / (5x) adalah ....

Learn how to solve limit problems involving trigonometric functions with small-angle approximations. This example covers two limit calculations: one with tan(x) and cos^2(x), and the other involving sin(3x) and cos(4x). Suitable for Grades 10-12 students.

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