If the function `f(x)=(sin3x+a sin 2x+b)/(x^(3)), x ne 0` is continuous at `x=0` and `f(0)=K, AA K in R`, then `b-a` is equal to

Options

A. 4
B. `(5)/(2)`
C. 5
D. `(3)/(2)`

Answer

To solve the problem step by step, we need to ensure that the function \( f(x) = \frac{\sin(3x) + a \sin(2x) + b}{x^3} \) is continuous at \( x = 0 \). This means that the limit of \( f(x) \) as \( x \) approaches 0 must equal \( f(0) = K \), where \( K \) is a finite number.

### Step 1: Find the limit as \( x \) approaches 0
We need to compute the limit:
\[ \lim_{x \to 0} f(x) = \lim_{x \to 0} \frac{\sin(3x) + a \sin(2x) + b}{x^3} \]

### Step 2: Use the Taylor series expansion for sine
The Taylor series expansion for \( \sin(x) \) around \( x = 0 \) is:
\[ \sin(x) = x - \frac{x^3}{6} + O(x^5) \]
Using this, we can expand \( \sin(3x) \) and \( \sin(2x) \):
\[ \sin(3x) = 3x - \frac{(3x)^3}{6} + O(x^5) = 3x - \frac{27x^3}{6} + O(x^5) = 3x - \frac{27}{6}x^3 + O(x^5) \]
\[ \sin(2x) = 2x - \frac{(2x)^3}{6} + O(x^5) = 2x - \frac{8x^3}{6} + O(x^5) = 2x - \frac{4}{3}x^3 + O(x^5) \]

### Step 3: Substitute the expansions into \( f(x) \)
Now substituting these expansions back into \( f(x) \):
\[ f(x) = \frac{(3x - \frac{27}{6}x^3) + a(2x - \frac{4}{3}x^3) + b}{x^3} \]
This simplifies to:
\[ f(x) = \frac{(3 + 2a)x - \left(\frac{27}{6} + \frac{4a}{3}\right)x^3 + b}{x^3} \]

### Step 4: Separate terms
Now we can separate the terms:
\[ f(x) = \frac{(3 + 2a)}{x^2} + \left(b - \frac{27}{6} - \frac{4a}{3}\right) \]

### Step 5: Ensure the limit exists
For \( f(x) \) to be continuous at \( x = 0 \), the term \( \frac{(3 + 2a)}{x^2} \) must vanish as \( x \) approaches 0. This implies:
\[ 3 + 2a = 0 \implies a = -\frac{3}{2} \]

### Step 6: Solve for \( b \)
Next, we set the constant term to be finite:
\[ b - \frac{27}{6} - \frac{4a}{3} = 0 \]
Substituting \( a = -\frac{3}{2} \):
\[ b - \frac{27}{6} - \frac{4(-\frac{3}{2})}{3} = 0 \]
\[ b - \frac{27}{6} + 2 = 0 \]
\[ b = \frac{27}{6} - 2 = \frac{27}{6} - \frac{12}{6} = \frac{15}{6} = \frac{5}{2} \]

### Step 7: Calculate \( b - a \)
Now we can find \( b - a \):
\[ b - a = \frac{5}{2} - \left(-\frac{3}{2}\right) = \frac{5}{2} + \frac{3}{2} = \frac{8}{2} = 4 \]

### Final Answer
Thus, \( b - a = 4 \).