The value of `lim_(x to 0) ("sin" alpha X - "sin" beta x)/(e^(alphax) - e^(beta x))` equals

Options

A. 0
B. 1
C. `-1`
D. `alpha - beta`

Answer

To solve the limit problem

\[ \lim_{x \to 0} \frac{\sin(\alpha x) - \sin(\beta x)}{e^{\alpha x} - e^{\beta x}}, \]

we will follow these steps:

### Step 1: Rewrite the limit
We can rewrite the limit by adding and subtracting 1 in the denominator:

\[ \lim_{x \to 0} \frac{\sin(\alpha x) - \sin(\beta x)}{(e^{\alpha x} - 1) - (e^{\beta x} - 1)}. \]

### Step 2: Factor the denominator
Now we can factor the denominator:

\[ = \lim_{x \to 0} \frac{\sin(\alpha x) - \sin(\beta x)}{e^{\alpha x} - 1 - (e^{\beta x} - 1)}. \]

### Step 3: Divide numerator and denominator by \(x\)
Next, we will divide both the numerator and denominator by \(x\):

\[ = \lim_{x \to 0} \frac{\frac{\sin(\alpha x) - \sin(\beta x)}{x}}{\frac{e^{\alpha x} - 1}{x} - \frac{e^{\beta x} - 1}{x}}. \]

### Step 4: Apply limit properties
Using the known limits:

1. \(\lim_{x \to 0} \frac{\sin(kx)}{x} = k\) for any constant \(k\),
2. \(\lim_{x \to 0} \frac{e^{kx} - 1}{x} = k\) for any constant \(k\),

we can evaluate the limits:

\[ = \frac{\alpha - \beta}{\alpha - \beta} = 1. \]

### Final Answer
Thus, the value of the limit is

\[ \boxed{1}. \]