If `lim_(x rarr 0)x[4/x]= A`, then the value of x at which `f(x) = [x^2]sinpix` is discontinuous , (where `[.]` denotes greatest integer function)

Options

A. `sqrt(A+1)`
B. `sqrt(A+5)`
C. `sqrt(A)`
D. `sqrt(A+21)`

Answer

To solve the problem, we need to analyze the limit and the function given in the question step by step.

### Step 1: Evaluate the limit
We start with the limit:
\[ \lim_{x \to 0} x \left[ \frac{4}{x} \right] \]
Here, \([ \cdot ]\) denotes the greatest integer function. We can rewrite this as:
\[ \lim_{x \to 0} x \left( \frac{4}{x} - \left\{ \frac{4}{x} \right\} \right) \]
where \(\{ \cdot \}\) denotes the fractional part of the number.

### Step 2: Simplify the limit expression
This can be simplified to:
\[ \lim_{x \to 0} (4 - x \left\{ \frac{4}{x} \right\}) \]
As \(x\) approaches 0, \(\left\{ \frac{4}{x} \right\}\) oscillates between 0 and 1, but it remains bounded. Thus, \(x \left\{ \frac{4}{x} \right\}\) approaches 0.

### Step 3: Calculate the limit
Therefore, we have:
\[ \lim_{x \to 0} (4 - x \left\{ \frac{4}{x} \right\}) = 4 - 0 = 4 \]
Thus, we find that:
\[ A = 4 \]

### Step 4: Analyze the function \(f(x)\)
Now we need to find the points of discontinuity of the function:
\[ f(x) = \left[ x^2 \sin(\pi x) \right] \]
The function \(\left[ x^2 \sin(\pi x) \right]\) will be discontinuous at points where \(x^2 \sin(\pi x)\) is an integer.

### Step 5: Identify points of discontinuity
The function \(\sin(\pi x)\) is 0 at integer values of \(x\). Therefore, \(f(x)\) will be discontinuous at integer values of \(x\) because:
- At integers, \(x^2 \sin(\pi x) = 0\), which is an integer.
- The greatest integer function \([x^2 \sin(\pi x)]\) will jump at these points.

### Step 6: Check for irrational points
Additionally, \(f(x)\) could also be discontinuous at points where \(x^2\) is an integer (i.e., \(x = \sqrt{n}\) for integers \(n\)). However, since we are primarily looking for discontinuities related to the greatest integer function, we focus on the integers.

### Conclusion
The function \(f(x)\) is discontinuous at all integer values of \(x\).