If `f(x)={alpha+sin[x]/x , x > 0 and 2 ,x=0 and beta+[(sin x-x)/x^3] ,x < 0` (whlenotes the greatest integer function) if `f(x)` is continuous at `x = 0`. then `beta` is equal to

Options

A. `alpha-1`
B. `alpha+1`
C. `alpha+2`
D. `alpha-2`

Explanation

To solve the problem, we need to ensure that the function \( f(x) \) is continuous at \( x = 0 \). This means that the left-hand limit, the right-hand limit, and the value of the function at that point must all be equal.

The function is defined as follows:
- For \( x > 0 \): \( f(x) = \alpha + \frac{\sin(\lfloor x \rfloor)}{x} \)
- For \( x = 0 \): \( f(0) = 2 \)
- For \( x < 0 \): \( f(x) = \beta + \frac{\sin x - x}{x^3} \)

### Step 1: Calculate the Right-Hand Limit as \( x \to 0^+ \)

For \( x > 0 \):
\[ f(x) = \alpha + \frac{\sin(\lfloor x \rfloor)}{x} \]
As \( x \to 0^+ \), \( \lfloor x \rfloor = 0 \), so:
\[ f(x) = \alpha + \frac{\sin(0)}{x} = \alpha + 0 = \alpha \]
Thus, the right-hand limit is:
\[ \lim_{x \to 0^+} f(x) = \alpha \]

### Step 2: Calculate the Left-Hand Limit as \( x \to 0^- \)

For \( x < 0 \):
\[ f(x) = \beta + \frac{\sin x - x}{x^3} \]
We need to evaluate:
\[ \lim_{x \to 0^-} f(x) = \beta + \lim_{x \to 0^-} \frac{\sin x - x}{x^3} \]
Using the Taylor series expansion for \( \sin x \):
\[ \sin x = x - \frac{x^3}{6} + O(x^5) \]
Thus:
\[ \sin x - x = -\frac{x^3}{6} + O(x^5) \]
So:
\[ \frac{\sin x - x}{x^3} = \frac{-\frac{x^3}{6} + O(x^5)}{x^3} = -\frac{1}{6} + O(x^2) \]
As \( x \to 0 \), the limit becomes:
\[ \lim_{x \to 0^-} \frac{\sin x - x}{x^3} = -\frac{1}{6} \]
Thus:
\[ \lim_{x \to 0^-} f(x) = \beta - \frac{1}{6} \]

### Step 3: Set the Limits Equal to Ensure Continuity

For continuity at \( x = 0 \):
\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^-} f(x) = f(0) \]
This gives us the equation:
\[ \alpha = \beta - \frac{1}{6} = 2 \]

### Step 4: Solve for \( \beta \)

From \( \alpha = 2 \):
\[ \beta - \frac{1}{6} = 2 \]
Adding \( \frac{1}{6} \) to both sides:
\[ \beta = 2 + \frac{1}{6} = \frac{12}{6} + \frac{1}{6} = \frac{13}{6} \]

### Final Answer
Thus, the value of \( \beta \) is:
\[ \beta = \frac{13}{6} \]