If f(x) = [x] – [x/4] , x ∈ R, where [x] denotes the greatest integer function, then :
(1) lim f(x) (x→4-)exists but lim f(x) (x→4+) does not exist.
(2) Both lim f(x) (x→4-) and lim f(x) (x→4+) exist but are not equal.
(3) lim f(x) (x→4+) exists but lim f(x) (x→4-) does not exist.
(4) f is continuous at x = 4.

Answer

To solve the problem, we need to analyze the function \( f(x) = [x] - \left[\frac{x}{4}\right] \) as \( x \) approaches 4 from both the left and the right.

### Step 1: Calculate \( \lim_{x \to 4^-} f(x) \)

When \( x \) approaches 4 from the left (i.e., \( x \to 4^- \)), we can consider values like \( 3.9, 3.99, \) etc.

1. **Evaluate \( [x] \)**:
- For \( x < 4 \), \( [x] = 3 \).

2. **Evaluate \( \left[\frac{x}{4}\right] \)**:
- For \( x < 4 \), \( \frac{x}{4} < 1 \), hence \( \left[\frac{x}{4}\right] = 0 \).

Now substituting these values into the function:
\[ f(x) = [x] - \left[\frac{x}{4}\right] = 3 - 0 = 3. \]

Thus,
\[ \lim_{x \to 4^-} f(x) = 3. \]

### Step 2: Calculate \( \lim_{x \to 4^+} f(x) \)

When \( x \) approaches 4 from the right (i.e., \( x \to 4^+ \)), we can consider values like \( 4.1, 4.01, \) etc.

1. **Evaluate \( [x] \)**:
- For \( x \geq 4 \), \( [x] = 4 \).

2. **Evaluate \( \left[\frac{x}{4}\right] \)**:
- For \( x \geq 4 \), \( \frac{x}{4} \geq 1 \), hence \( \left[\frac{x}{4}\right] = 1 \).

Now substituting these values into the function:
\[ f(x) = [x] - \left[\frac{x}{4}\right] = 4 - 1 = 3. \]

Thus,
\[ \lim_{x \to 4^+} f(x) = 3. \]

### Step 3: Evaluate \( f(4) \)

Now we need to find the value of the function at \( x = 4 \):
\[ f(4) = [4] - \left[\frac{4}{4}\right] = 4 - 1 = 3. \]

### Conclusion

We have:
\[ \lim_{x \to 4^-} f(x) = 3, \quad \lim_{x \to 4^+} f(x) = 3, \quad \text{and } f(4) = 3. \]

Since both one-sided limits exist and are equal to the value of the function at that point, we conclude that \( f \) is continuous at \( x = 4 \).

### Final Answer

The correct option is:
(4) \( f \) is continuous at \( x = 4 \).

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