Let `lim_(x to 0) ("sin" 2X)/(x) = a` and `lim_(x to 0) (3x)/(tan x) = b`, then a + b equals

Options

A. 5
B. 6
C. 0
D. 4

Answer

To solve the problem, we need to evaluate the limits given in the question and then find the sum of the results.

### Step-by-Step Solution

1. **Evaluate \( a = \lim_{x \to 0} \frac{\sin(2x)}{x} \)**:
- We can use the property of limits that states \( \lim_{x \to 0} \frac{\sin(kx)}{x} = k \) for any constant \( k \).
- Here, \( k = 2 \).
- Therefore, we can rewrite the limit as:
\[ a = \lim_{x \to 0} \frac{\sin(2x)}{x} = \lim_{x \to 0} \frac{\sin(2x)}{2x} \cdot 2 = 1 \cdot 2 = 2 \]

2. **Evaluate \( b = \lim_{x \to 0} \frac{3x}{\tan(x)} \)**:
- We can use the property that \( \lim_{x \to 0} \frac{\tan(x)}{x} = 1 \).
- Thus, we can rewrite the limit as:
\[ b = \lim_{x \to 0} \frac{3x}{\tan(x)} = 3 \cdot \lim_{x \to 0} \frac{x}{\tan(x)} = 3 \cdot \frac{1}{1} = 3 \]

3. **Calculate \( a + b \)**:
- Now that we have \( a = 2 \) and \( b = 3 \), we can find their sum:
\[ a + b = 2 + 3 = 5 \]

### Final Answer
Thus, \( a + b = 5 \).