https://file.aiquickdraw.com/mathbot/file/1726698061286dstivtm8.jpg

The expression in the image is a limit problem:

\[
\lim_{{x \to 0}} \frac{\tan^3(2x)}{x^2 \sin(7x)}.
\]

Let's go through the steps to solve this limit.

### Step-by-step approach:

1. **Simplify using small-angle approximations**:
   As \(x \to 0\), both \(\tan(2x)\) and \(\sin(7x)\) can be approximated by their respective small-angle approximations:
   - \(\tan(2x) \approx 2x\),
   - \(\sin(7x) \approx 7x\).

2. **Substitute these approximations**:
   Using these approximations, the limit becomes:
   \[
   \lim_{{x \to 0}} \frac{(2x)^3}{x^2 \cdot 7x} = \lim_{{x \to 0}} \frac{8x^3}{7x^3}.
   \]

3. **Simplify the expression**:
   The \(x^3\) terms cancel out, leaving:
   \[
   \frac{8}{7}.
   \]

### Final Answer:
\[
\lim_{{x \to 0}} \frac{\tan^3(2x)}{x^2 \sin(7x)} = \frac{8}{7}.
\]

Would you like a more detailed explanation of any of the steps, or have any questions about this process?

Here are 5 related questions for further practice:
1. How would you solve \(\lim_{{x \to 0}} \frac{\sin(3x)}{x^3}\)?
2. What is the limit of \(\lim_{{x \to 0}} \frac{\tan(x)}{\sin(x)}\)?
3. How do you apply L'Hopital's Rule to a limit involving indeterminate forms like \(\frac{0}{0}\)?
4. Can you compute \(\lim_{{x \to 0}} \frac{x^2}{\sin(x)}\)?
5. What is the small-angle approximation for \(\cos(x)\), and how does it affect limits?

### Tip:
Always check if a limit leads to an indeterminate form such as \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), as these indicate the potential use of L'Hopital's Rule or small-angle approximations.

Find the limit of tan^3(2x) / (x^2 * sin(7x)) as x approaches 0.

Learn how to find the limit of tan^3(2x) / (x^2 * sin(7x)) as x approaches 0 using small-angle approximations. This problem covers trigonometric functions and limits. Suitable for calculus students in Grades 11-12.

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