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.