`lim _(xtooo)3/x [(x)/(4)]=p/q` where [.] denotes greatest integer function), then `p+q` (where p,q are relative prime) is:

Options

A. 2
B. 7
C. 5
D. 6

Answer

To solve the limit problem step by step, we start with the expression:

\[ \lim_{x \to \infty} \frac{3}{x} \left[ \frac{x}{4} \right] \]

where \([.]\) denotes the greatest integer function.

### Step 1: Understanding the Greatest Integer Function
The greatest integer function \([x]\) gives the largest integer less than or equal to \(x\). For our case, we need to evaluate \(\left[ \frac{x}{4} \right]\).

### Step 2: Rewrite the Expression
We can express \(\left[ \frac{x}{4} \right]\) as:
\[ \left[ \frac{x}{4} \right] = \frac{x}{4} - \left\{ \frac{x}{4} \right\} \]
where \(\left\{ \frac{x}{4} \right\}\) is the fractional part of \(\frac{x}{4}\).

### Step 3: Substitute into the Limit
Substituting this into our limit gives:
\[ \lim_{x \to \infty} \frac{3}{x} \left( \frac{x}{4} - \left\{ \frac{x}{4} \right\} \right) \]

### Step 4: Distribute the Terms
Distributing the \(\frac{3}{x}\) into the expression:
\[ \lim_{x \to \infty} \left( \frac{3}{x} \cdot \frac{x}{4} - \frac{3}{x} \left\{ \frac{x}{4} \right\} \right) \]

This simplifies to:
\[ \lim_{x \to \infty} \left( \frac{3}{4} - \frac{3}{x} \left\{ \frac{x}{4} \right\} \right) \]

### Step 5: Analyze the Fractional Part
As \(x \to \infty\), the fractional part \(\left\{ \frac{x}{4} \right\}\) will always be between \(0\) and \(1\). Thus, \(\frac{3}{x} \left\{ \frac{x}{4} \right\}\) approaches \(0\) because \(\frac{3}{x}\) approaches \(0\).

### Step 6: Evaluate the Limit
Now we can evaluate the limit:
\[ \lim_{x \to \infty} \left( \frac{3}{4} - 0 \right) = \frac{3}{4} \]

### Step 7: Express as a Fraction
We have:
\[ \frac{3}{4} = \frac{p}{q} \]
where \(p = 3\) and \(q = 4\).

### Step 8: Find \(p + q\)
Since \(p\) and \(q\) are relatively prime, we calculate:
\[ p + q = 3 + 4 = 7 \]

Thus, the final answer is:
\[ \boxed{7} \]