https://file.aiquickdraw.com/mathbot/file/1727117752230niq759s0.jpg

I see you've uploaded an image with some mathematical expressions and limits. Let's analyze it:

From the top section of the image, I can see a limit expression involving trigonometric functions and the limit as \( x \to 0 \):

\[
\lim_{x \to 0} \frac{7}{2} \cdot \frac{\sin(3x)}{3x} \cdot \frac{1}{\cos(6x)} = \frac{7}{2}
\]

This appears to be using a known limit identity:
\[
\lim_{x \to 0} \frac{\sin(kx)}{kx} = 1
\]
For \( k = 3 \), \( \lim_{x \to 0} \frac{\sin(3x)}{3x} = 1 \), and as \( \cos(6x) \to 1 \) when \( x \to 0 \), the entire expression simplifies to \( \frac{7}{2} \).

The second part of the image focuses on:
\[
\lim_{x \to 0} \frac{\tan^3(2x)}{x^2 \sin(3x)}
\]

Using approximations for small \( x \):
- \( \tan(2x) \approx 2x \)
- \( \sin(3x) \approx 3x \)

Substituting, this becomes:
\[
\lim_{x \to 0} \frac{(2x)^3}{x^2 (3x)} = \lim_{x \to 0} \frac{8x^3}{3x^3} = \frac{8}{3}
\]

Thus, the limit simplifies to \( \frac{8}{3} \).

Would you like a detailed breakdown of the steps or any clarification on these limits?

### Five Related Questions:
1. What are the standard limit identities for trigonometric functions?
2. How can L'Hopital's Rule be applied to limits like these?
3. What is the small angle approximation for \( \sin(x) \) and \( \tan(x) \)?
4. How does the cosine function behave near zero?
5. What are common trigonometric limits involving powers of \( x \)?

### Tip:
When evaluating limits involving trigonometric functions, especially as \( x \to 0 \), applying small-angle approximations like \( \sin(x) \approx x \) and \( \tan(x) \approx x \) often simplifies the process significantly.

Evaluate the limits involving trigonometric functions, specifically lim x->0 of (7/2) * (sin(3x)/3x) * (1/cos(6x)) and lim x->0 of (tan^3(2x))/(x^2 * sin(3x))

Learn how to evaluate trigonometric limits using small angle approximations and limit theorems. This problem covers limits involving sin(x), cos(x), and tan(x) near x -> 0, using known trigonometric identities.

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