lim
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3 ∙ 𝑠𝑒𝑛(𝑥𝑦)
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Let's evaluate the limit as \((x, y) \to (0, 0)\) of the expression:

\[
\lim_{(x, y) \to (0, 0)} \frac{3 \cdot \sin(xy)}{x}
\]

### Step 1: Analyze the expression
The expression given is \(\frac{3 \cdot \sin(xy)}{x}\).

### Step 2: Consider the behavior of \(\sin(xy)\)
For small values of \(xy\), we can use the approximation:

\[
\sin(xy) \approx xy \quad \text{(as \(xy \to 0\))}
\]

So, the expression becomes:

\[
\frac{3 \cdot \sin(xy)}{x} \approx \frac{3 \cdot xy}{x} = 3y
\]

### Step 3: Evaluate the limit
Now, the limit simplifies to:

\[
\lim_{(x, y) \to (0, 0)} 3y
\]

As \((x, y) \to (0, 0)\), \(y\) approaches 0. Therefore:

\[
\lim_{(x, y) \to (0, 0)} 3y = 3 \cdot 0 = 0
\]

### Final Answer:
\[
\lim_{(x, y) \to (0, 0)} \frac{3 \cdot \sin(xy)}{x} = 0
\]

Would you like more details on this process, or do you have any questions?

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Here are some related questions:

1. How do you apply Taylor series to approximate trigonometric functions near 0?
2. What is the importance of limits in understanding the behavior of multivariable functions?
3. How do you handle limits that involve trigonometric functions?
4. Can you explain the small angle approximation in trigonometry?
5. What are the differences between single-variable and multivariable limits?

**Tip:** When dealing with limits involving trigonometric functions, using approximations for small angles (like \(\sin(x) \approx x\) near 0) can simplify the calculation significantly.

Learn how to evaluate the limit \(\lim_{(x, y) \to (0, 0)} \frac{3 \cdot \sin(xy)}{x}\) using trigonometric approximations and the squeeze theorem. Understand the behavior of multivariable functions near the origin. Suitable for advanced undergraduate students studying multivariable calculus.

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