Evaluate the limit [[2x - x²]] / [[2 - x]] as x→2⁺

limit [[2x - x²]] / [[2 - x]] x→2⁺

My answer sheet says the following:

x > 2
2-x < 0
therefore [[2 - x]] = -1

2x-x^2
x(2-x) = approximate 0
(>2)(<0) < 0

[[2x - x^2]] = -1

so

[[2x-x^2]]/[[2-x]] = (-1/-1) = 1

But how? I still don't get it!

Thank you very much!

[[ x ]] means greatest integer less than or equal to x:

Examples:
[[ 4 ]] = 4
[[ 2.5 ]] = 2
[[ π ]] = 3
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Let's try a few numbers to the right of x = 2:
x = 2.1 ==> [[2(2.1) - (2.1)^2]] / [[2 - 2.1]] = [-0.21] / [-0.1] = (-1)/(-1) = 1.
x = 2.01 ==> [[2(2.01) - (2.01)^2]] / [[2 - 2.01]] = [-0.0201] / [-0.01] = (-1)/(-1) = 1.
and so on.

So, it seems that the limit equals 1.
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Showing this algebraically is a little trickier, being of the form 0/0 and with double brackets.

Note that for x → 2+ (think something along the lines of x = 2.0001),

(i) 2 - x is a real number strictly between -1 and 0. Hence, [[2 - x]] = -1.

(ii) 2x - x^2 = x(2 - x) will be also between -1 and 0. Hence, [[2x - x^2]] = -1 as well.

Hence, the limit equals (-1) / (-1) = 1.
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More algebraically (read on, if you must...):
Choose x sufficiently close to 2 from the right: 2 < x < 2 + ε, for sufficiently small ε, say less than 1/1000 (to be safe).

So, -(2 + ε) < -x < -2
==> -ε < 2 - x < 0.

Since -1/1000 < -ε < 0, we have [[ 2 - x ]] = -1.
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Next, multiply both sides of -ε < 2 - x < 0 by x > 0:
-εx < x(2 - x) < 0

Since -(2 + ε) < -x < -2, we can rewrite this as
-(2 + ε)ε < -x * ε < x(2 - x) < 0

Since we chose ε < 1/1000, we have -(2 + ε)ε > -(2 + .001) * .001 = -0.00201.

Placing these inequalities together, -0.00201 < -(2 + ε)ε < -x * ε < x(2 - x) < 0
==> -0.00201 < x(2 - x) < 0
==> [[ 2x - x^2 ]] = -1, as well.

Now, the limit proceeds as before: (-1)/(-1) = 1.
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I hope this helps!

I suppose that it's asking for absolute value? Well, let's first ask what the limit of (2x - x^2) / (2 - x) is as x goes to 2

2x - x^2 = x * (2 - x)

x * (2 - x) / (2 - x) =>
x

x goes to 2

So the limit goes to 2. I don't get how they're figuring that it goes to 1. That doesn't make any sense.