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We are tasked with evaluating the following limit:

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

Let's solve this step by step.

### Step 1: Apply small-angle approximations
For small \(x\), we can use the following approximations for \(\tan(x)\) and \(\sin(x)\):

- \(\tan(x) \approx x\) when \(x\) is close to 0.
- \(\sin(x) \approx x\) when \(x\) is close to 0.

Thus:
- \(\tan(3x^2) \approx 3x^2\) for small \(x\),
- \(\sin(7x) \approx 7x\), and therefore \(\sin^2(7x) \approx (7x)^2 = 49x^2\).

### Step 2: Substitute these approximations
Substitute these small-angle approximations into the given expression:

\[
\tan(3x^2) + \sin^2(7x) \approx 3x^2 + 49x^2 = 52x^2
\]

Thus, the limit becomes:

\[
\lim_{x \to 0} \frac{52x^2}{x^2}
\]

### Step 3: Simplify the expression
Simplifying the above expression:

\[
\frac{52x^2}{x^2} = 52
\]

### Final Answer:
\[
\lim_{x \to 0} \frac{\tan(3x^2) + \sin^2(7x)}{x^2} = 52
\]

If you have any questions or want more details, feel free to ask!

### Related Questions:
1. How do you apply L'Hôpital's Rule to limits involving indeterminate forms?
2. What are the common small-angle approximations for trigonometric functions?
3. How can you determine if a limit results in an indeterminate form?
4. What happens if the angle in \(\tan(x)\) or \(\sin(x)\) isn't small enough to use the approximation?
5. What other techniques can be used for evaluating limits as \(x \to 0\)?

### Tip:
Always remember that small-angle approximations like \(\sin(x) \approx x\) and \(\tan(x) \approx x\) are valid when \(x\) is close to 0, and can significantly simplify limit problems.

Find the limit: lim (x → 0) [(tan(3x²) + sin²(7x)) / x²]

Explore how to solve the limit lim (x → 0) [(tan(3x²) + sin²(7x)) / x²] using small-angle approximations. This detailed solution covers key concepts in calculus and trigonometry, perfect for high school and college-level students.

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