`lim_(x to 0) (2x)/(tan3x)=?`

Options

A. 3
B. 2
C. `(2)/(3)`
D. `(3)/(2)`

Answer

To solve the limit \( \lim_{x \to 0} \frac{2x}{\tan(3x)} \), we will follow these steps:

### Step 1: Identify the Form
First, we substitute \( x = 0 \) into the expression:
\[ \frac{2(0)}{\tan(3 \cdot 0)} = \frac{0}{0} \]
This is an indeterminate form, so we need to apply other techniques to evaluate the limit.

**Hint:** When you encounter a \( \frac{0}{0} \) form, consider using L'Hôpital's Rule or algebraic manipulation.

### Step 2: Rewrite the Limit
We can rewrite the limit as:
\[ \lim_{x \to 0} \frac{2x}{\tan(3x)} = \lim_{x \to 0} \frac{2x}{3x} \cdot \frac{3x}{\tan(3x)} \]
This allows us to separate the limit into two parts.

**Hint:** Factor out constants and rewrite the limit to make use of known limits.

### Step 3: Evaluate the Known Limit
We know from calculus that:
\[ \lim_{u \to 0} \frac{\tan(u)}{u} = 1 \]
Thus, we can apply this limit to our expression:
\[ \lim_{x \to 0} \frac{3x}{\tan(3x)} = \frac{1}{3} \]
This means:
\[ \lim_{x \to 0} \frac{3x}{\tan(3x)} = 1 \implies \lim_{x \to 0} \frac{\tan(3x)}{3x} = 1 \]

**Hint:** Remember the limit property of \( \tan(x) \) as \( x \) approaches 0.

### Step 4: Combine the Limits
Now we can combine the limits:
\[ \lim_{x \to 0} \frac{2x}{\tan(3x)} = \lim_{x \to 0} \frac{2x}{3x} \cdot \lim_{x \to 0} \frac{3x}{\tan(3x)} = \frac{2}{3} \cdot 1 \]

### Step 5: Final Result
Thus, we find:
\[ \lim_{x \to 0} \frac{2x}{\tan(3x)} = \frac{2}{3} \]

### Conclusion
The final answer is:
\[ \boxed{\frac{2}{3}} \]

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