To find the limit of the given expression, we will use the squeeze theorem.
The squeeze theorem states that if we have three functions, f(x), g(x), and h(x), such that g(x) ≤ f(x) ≤ h(x) for all x close to a certain value a (except possibly at x=a), and if Lim[x→a] g(x) = Lim[x→a] h(x) = L, then Lim[x→a] f(x) also equals L.
Let's start by finding the limits of the upper and lower bounds.
1. Upper bound: h(x) = x^2
To find Lim[x→+inf] x^2, we can see that as x approaches infinity, the function x^2 also approaches infinity. Therefore, Lim[x→+inf] x^2 = +inf.
2. Lower bound: g(x) = [[x^2]]
[[x^2]] is the greatest integer function, which means it rounds down any input to the nearest integer. Since x^2 is a positive number and [[x^2]] rounds it down, [[x^2]] will always be less than or equal to x^2. Therefore, g(x) = [[x^2]] ≤ x^2 for all x.
Now, using the squeeze theorem, we have:
Lim[x→+inf] g(x) ≤ Lim[x→+inf] f(x) ≤ Lim[x→+inf] h(x)
Lim[x→+inf] [[x^2]] ≤ Lim[x→+inf] (x^2 - [[x^2]])/2 ≤ Lim[x→+inf] x^2
Since Lim[x→+inf] [[x^2]] = +inf (as explained above), and Lim[x→+inf] x^2 = +inf, we can conclude that:
+inf ≤ Lim[x→+inf] (x^2 - [[x^2]])/2 ≤ +inf
Hence, by the squeeze theorem, Lim[x→+inf] (x^2 - [[x^2]])/2 = +inf.
Therefore, the limit of the given expression as x approaches positive infinity is +inf.