`lim_(x to 0) (x tan 3 x)/("sin"^(2) x)` is

Options

A. 0
B. 1
C. 2
D. 3

Answer

To evaluate the limit \( \lim_{x \to 0} \frac{x \tan(3x)}{\sin^2(x)} \), we can follow these steps:

### Step 1: Rewrite the limit expression
We start with the limit expression:
\[ \lim_{x \to 0} \frac{x \tan(3x)}{\sin^2(x)} \]

### Step 2: Manipulate the expression
We can rewrite the expression by multiplying and dividing by \( x^2 \):
\[ \lim_{x \to 0} \frac{x \tan(3x)}{\sin^2(x)} = \lim_{x \to 0} \frac{x \tan(3x)}{x^2} \cdot \frac{x^2}{\sin^2(x)} \]

### Step 3: Split the limit
Now we can split the limit into two parts:
\[ = \lim_{x \to 0} \frac{\tan(3x)}{x} \cdot \lim_{x \to 0} \frac{x^2}{\sin^2(x)} \]

### Step 4: Evaluate the first limit
Using the limit property \( \lim_{x \to 0} \frac{\tan(kx)}{kx} = 1 \), we can evaluate the first limit:
\[ \lim_{x \to 0} \frac{\tan(3x)}{x} = 3 \quad \text{(because we multiply by 3)} \]

### Step 5: Evaluate the second limit
For the second limit, we use the property \( \lim_{x \to 0} \frac{\sin(x)}{x} = 1 \):
\[ \lim_{x \to 0} \frac{x^2}{\sin^2(x)} = \left( \lim_{x \to 0} \frac{x}{\sin(x)} \right)^2 = 1^2 = 1 \]

### Step 6: Combine the results
Now we can combine the results from both limits:
\[ \lim_{x \to 0} \frac{x \tan(3x)}{\sin^2(x)} = 3 \cdot 1 = 3 \]

### Final Answer
Thus, the limit is:
\[ \boxed{3} \]