(lim x --> +inf) of (x^2 - [[x^2]])/2

using Squeeze theorem only

thank you so much in advance! its for my midterm tonight!

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.