If f(x)=`{{:(a+(sin[x])/x,xgt0),(2,x=0),(b+[(sinx-x)/(x^(3))],","xgt0):}` (where [.] denotes the greatest integer function).
If f(x) is continuous at x=0 then a+b is equal to

Answer

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 (LHL) as \( x \) approaches 0 from the left must equal the right-hand limit (RHL) as \( x \) approaches 0 from the right, and both must equal \( f(0) \).

Given:

\[ f(x) = \begin{cases} a + \frac{\sin(\lfloor x \rfloor)}{x} & \text{if } x > 0 \\ 2 & \text{if } x = 0 \\ b + \left\lfloor \frac{\sin x - x}{x^3} \right\rfloor & \text{if } x < 0 \end{cases} \]

### Step 1: Calculate the Left-Hand Limit (LHL)

For \( x < 0 \):

\[ LHL = \lim_{x \to 0^-} \left( b + \left\lfloor \frac{\sin x - x}{x^3} \right\rfloor \right) \]

Using the Taylor series expansion for \( \sin x \):

\[ \sin x = x - \frac{x^3}{6} + O(x^5) \]

Thus,

\[ \sin x - x = -\frac{x^3}{6} + O(x^5) \]

Substituting this into our limit:

\[ \frac{\sin x - x}{x^3} = \frac{-\frac{x^3}{6} + O(x^5)}{x^3} = -\frac{1}{6} + O(x^2) \]

As \( x \to 0 \), the \( O(x^2) \) term approaches 0, so:

\[ \lim_{x \to 0^-} \frac{\sin x - x}{x^3} = -\frac{1}{6} \]

Thus, the greatest integer function gives:

\[ \left\lfloor -\frac{1}{6} \right\rfloor = -1 \]

So, we have:

\[ LHL = b - 1 \]

### Step 2: Calculate the Right-Hand Limit (RHL)

For \( x > 0 \):

\[ RHL = \lim_{x \to 0^+} \left( a + \frac{\sin(\lfloor x \rfloor)}{x} \right) \]

Since \( \lfloor x \rfloor = 0 \) as \( x \to 0^+ \):

\[ RHL = \lim_{x \to 0^+} \left( a + \frac{\sin(0)}{x} \right) = a + 0 = a \]

### Step 3: Set the Limits Equal to Each Other

For continuity at \( x = 0 \):

\[ LHL = RHL = f(0) \]

Thus:

\[ b - 1 = a \]

And since \( f(0) = 2 \):

\[ b - 1 = 2 \implies b = 3 \]

### Step 4: Substitute \( b \) into the Equation

From \( b - 1 = a \):

\[ 3 - 1 = a \implies a = 2 \]

### Step 5: Calculate \( a + b \)

Now, we can find \( a + b \):

\[ a + b = 1 + 3 = 4 \]

### Final Answer

Thus, the value of \( a + b \) is:

\[ \boxed{4} \]