If `[.]` denotes the greatest integer function then `lim_(x->oo)([x]+[2x]+[3x]+[4x])/x^2` is

Options

A. 0
B. x
C. `x/2`
D. `(x^(2))/2`

Answer

To solve the limit problem, we need to evaluate:

\[ \lim_{x \to \infty} \frac{[x] + [2x] + [3x] + [4x]}{x^2} \]

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

### Step 1: Rewrite the greatest integer functions
We can express each term involving the greatest integer function as follows:

\[ [x] = x - \{x\}, \quad [2x] = 2x - \{2x\}, \quad [3x] = 3x - \{3x\}, \quad [4x] = 4x - \{4x\} \]

where \(\{y\}\) denotes the fractional part of \(y\).

### Step 2: Substitute into the limit expression
Substituting these expressions into our limit gives:

\[ [x] + [2x] + [3x] + [4x] = (x - \{x\}) + (2x - \{2x\}) + (3x - \{3x\}) + (4x - \{4x\}) \]

This simplifies to:

\[ = (x + 2x + 3x + 4x) - (\{x\} + \{2x\} + \{3x\} + \{4x\}) = 10x - (\{x\} + \{2x\} + \{3x\} + \{4x\}) \]

### Step 3: Substitute back into the limit
Now, substituting this back into our limit expression gives:

\[ \lim_{x \to \infty} \frac{10x - (\{x\} + \{2x\} + \{3x\} + \{4x\})}{x^2} \]

### Step 4: Separate the limit
We can separate the limit into two parts:

\[ = \lim_{x \to \infty} \left( \frac{10x}{x^2} - \frac{\{x\} + \{2x\} + \{3x\} + \{4x\}}{x^2} \right) \]

### Step 5: Simplify each part
The first part simplifies to:

\[ \frac{10x}{x^2} = \frac{10}{x} \]

As \(x \to \infty\), this approaches \(0\).

The second part involves the sum of fractional parts:

\[ \frac{\{x\} + \{2x\} + \{3x\} + \{4x\}}{x^2} \]

Since each fractional part \(\{kx\}\) (for \(k = 1, 2, 3, 4\)) is bounded between \(0\) and \(1\), we can say:

\[ 0 \leq \{x\} + \{2x\} + \{3x\} + \{4x\} < 4 \]

Thus,

\[ 0 \leq \frac{\{x\} + \{2x\} + \{3x\} + \{4x\}}{x^2} < \frac{4}{x^2} \]

As \(x \to \infty\), \(\frac{4}{x^2} \to 0\).

### Step 6: Combine the results
Putting it all together, we have:

\[ \lim_{x \to \infty} \left( \frac{10}{x} - \frac{\{x\} + \{2x\} + \{3x\} + \{4x\}}{x^2} \right) = 0 - 0 = 0 \]

### Final Result
Thus, the limit is:

\[ \boxed{0} \]