$$ \lim_{r\rarr0}\frac{\text{ }\sin\left(r^3\cos^3\left(\theta\right)+r^2\sin^2\left(\theta\right)\right)}{r^2} $$

To evaluate the limit

\[
\lim_{r \to 0} \frac{\sin\left(r^3\cos^3(\theta) + r^2\sin^2(\theta)\right)}{r^2},
\]

we need to consider the behavior of the function as $r \to 0$. Let's break down the expression and analyze it step by step.

### Step 1: Approximation of the sine function for small arguments
Recall that for small $x$, $\sin(x)$ can be approximated by $x$, i.e., 
\[
\sin(x) \approx x \quad \text{as} \quad x \to 0.
\]
Thus, when $r$ is close to 0, we can approximate:
\[
\sin\left(r^3\cos^3(\theta) + r^2\sin^2(\theta)\right) \approx r^3\cos^3(\theta) + r^2\sin^2(\theta).
\]

### Step 2: Simplifying the limit expression
Now substitute the approximation into the original limit:
\[
\lim_{r \to 0} \frac{r^3\cos^3(\theta) + r^2\sin^2(\theta)}{r^2}.
\]

We can now separate the terms in the numerator:
\[
\lim_{r \to 0} \left( \frac{r^3\cos^3(\theta)}{r^2} + \frac{r^2\sin^2(\theta)}{r^2} \right).
\]
Simplifying each term:
\[
= \lim_{r \to 0} \left( r\cos^3(\theta) + \sin^2(\theta) \right).
\]

### Step 3: Evaluate the limit
As $r \to 0$, the term $r\cos^3(\theta)$ tends to 0. Therefore, the limit becomes:
\[
\lim_{r \to 0} \sin^2(\theta).
\]

Thus, the limit is:
\[
\boxed{\sin^2(\theta)}.
\]

---

Do you want more details on any step, or have any questions?

Here are 5 related questions to expand your understanding:
1. What is the general approximation for $\sin(x)$ near $x = 0$?
2. How does the small-angle approximation simplify trigonometric limits?
3. Can you explain why the higher-order terms vanish as $r \to 0$?
4. What would happen if we replaced $\sin(x)$ with $\cos(x)$ in the original problem?
5. How can we use Taylor series to refine the approximation in this limit?

**Tip:** When evaluating limits involving trigonometric functions, always check if small-angle approximations can simplify the expression, especially as the variable approaches zero.

$$ \lim_{r \to 0} \frac{\sin\left(r^3\cos^3\left(\theta\right) + r^2\sin^2\left(\theta\right)\right)}{r^2} $$

Learn how to evaluate the limit $$ \lim_{r \to 0} \frac{\sin\left(r^3\cos^3\left(\theta\right) + r^2\sin^2\left(\theta\right)\right)}{r^2} $$ using trigonometric approximations. This step-by-step solution covers key calculus concepts like limits and small-angle approximation, suitable for advanced high school or university-level students.

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