The value of \mathop {\lim }\limits_{n \to \infty } \frac{{[r] + [2r] + ... + [nr]}}{{{n^2}}},
where r is a non-zero real number and [r] denotes the greatest integer less than or equal to r, is equal to :

0

r

\frac{r}{2}

2r

We are tasked with finding the value of:

\[
\lim_{n \to \infty } \frac{[r] + [2r] + \dots + [nr]}{n^2},
\]
where \( r \) is a non-zero real number, and \([x]\) represents the greatest integer less than or equal to \(x\).

### Step 1: Analyze the summation

The summation in the numerator is:

\[
S_n = [r] + [2r] + [3r] + \dots + [nr].
\]

The term \([kr]\) refers to the greatest integer less than or equal to \(kr\), and for large \(n\), we need to understand how this sum behaves relative to \(n^2\).

### Step 2: Break down the sum

For large \(k\), the value \([kr]\) can be written as:

\[
[kr] = kr - \{kr\},
\]

where \(\{kr\}\) is the fractional part of \(kr\), which lies in the interval \([0, 1)\).

Thus, the sum becomes:

\[
S_n = (r + 2r + 3r + \dots + nr) - (\{r\} + \{2r\} + \{3r\} + \dots + \{nr\}).
\]

The first part of the sum is:

\[
r(1 + 2 + 3 + \dots + n) = r \cdot \frac{n(n+1)}{2} \sim \frac{r n^2}{2} \text{ for large } n.
\]

The second part is a sum of fractional parts, and since \(\{kr\} \in [0, 1)\), the sum \(\{r\} + \{2r\} + \dots + \{nr\}\) grows linearly with \(n\), i.e., it is approximately \(O(n)\).

### Step 3: Simplify the limit

Thus, for large \(n\), we have:

\[
S_n \sim \frac{r n^2}{2} - O(n).
\]

Dividing by \(n^2\), we get:

\[
\frac{S_n}{n^2} \sim \frac{r}{2} - \frac{O(n)}{n^2}.
\]

As \(n \to \infty\), the term \(\frac{O(n)}{n^2}\) vanishes, and we are left with:

\[
\lim_{n \to \infty} \frac{S_n}{n^2} = \frac{r}{2}.
\]

### Final Answer:

\[
\boxed{\frac{r}{2}}.
\]

Do you want further details or have any questions?

Here are 5 related questions to consider:
1. What happens if \( r \) is an integer in the given limit problem?
2. How does the behavior of the fractional part affect the limit?
3. Can this technique be applied to other sums involving greatest integer functions?
4. What if we replace \( n^2 \) with \( n^p \), where \( p \) is a different power?
5. How does this limit relate to the concept of Riemann sums in integral calculus?

**Tip:** For large sums involving step functions, consider separating the integer and fractional parts to simplify the analysis.

The value of \mathop{\lim }\limits_{n \to \infty } \frac{{[r] + [2r] + ... + [nr]}}{{{n^2}}}, where r is a non-zero real number and [r] denotes the greatest integer less than or equal to r, is equal to: 0, r, \frac{r}{2}, 2r

Explore the solution to the limit problem involving the greatest integer function: \lim_{n \to \infty} \frac{[r] + [2r] + ... + [nr]}{n^2}, where r is a real number. This detailed solution breaks down key concepts such as asymptotic behavior, the arithmetic series, and fractional parts. Suitable for undergraduate mathematics students.

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