Find the asymptotes of the curve `x y-3y-2x=0` .

Explanation

To find the asymptotes of the curve given by the equation \( xy - 3y - 2x = 0 \), we can follow these steps:

### Step 1: Rearranging the Equation
Start by rearranging the given equation to isolate the terms involving \( y \):
\[ xy - 3y = 2x \]

### Step 2: Factoring Out \( y \)
Factor \( y \) out from the left side:
\[ y(x - 3) = 2x \]

### Step 3: Solving for \( y \)
Now, solve for \( y \):
\[ y = \frac{2x}{x - 3} \]

### Step 4: Identifying Asymptotes
To find the asymptotes, we need to analyze the behavior of \( y \) as \( x \) approaches certain values. The vertical asymptote occurs where the denominator is zero:
\[ x - 3 = 0 \implies x = 3 \]

Next, we find the horizontal asymptote by considering the limit of \( y \) as \( x \) approaches infinity:
\[ \lim_{x \to \infty} y = \lim_{x \to \infty} \frac{2x}{x - 3} = \lim_{x \to \infty} \frac{2}{1 - \frac{3}{x}} = 2 \]

### Step 5: Writing the Asymptotes
Thus, we have the following asymptotes:
1. Vertical asymptote: \( x = 3 \)
2. Horizontal asymptote: \( y = 2 \)

### Final Answer
The asymptotes of the curve \( xy - 3y - 2x = 0 \) are:
- Vertical: \( x = 3 \)
- Horizontal: \( y = 2 \)

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