Calculus HELP!!!!!?

Question

How do you do step (greatest integer/least integer) functions with theta or with transformations? Ex:
The limit as x goes to 3+ of the greatest integer [θ]/θ. I know the answer is 1 but I'm not sure how.
OR: the limit as x goes to 4+ of x- the greatest integer x:
(x-[x])

I hope you understand the question. I don't have the calc symbols on my keyboard.

Answer 1

Well, do you know how to find one-sided limits of the floor and ceiling functions? If not, remember that the floor function is right continuous, so [x→c⁺]lim ⌊x⌋ = ⌊c⌋ for any c. Similarly, the floor function is left-continuous at every point except the integers, where it has a jump discontinuity of size 1. So [x→c⁻]lim ⌊c⌋ = {c-1 if c∈ℤ, ⌊c⌋ otherwise).

Now, when finding limits of composite functions, remember that [x→c⁺]lim (f(x)/g(x)) = ([x→c⁺]lim f(x)) / ([x→c⁺]lim g(x)) provided both limits exist and are finite and [x→c⁺]lim g(x) ≠ 0 (with suitable definitions of multiplication and division by infinity, you can weaken the first condition to one limit being finite). The same thing holds true for left hand limits. So you can just take the limits one term at a time:

[θ→3⁺]lim ⌊θ⌋/θ
([θ→3⁺]lim ⌊θ⌋) / ([θ→3⁺]lim θ)

Now, [θ→3⁺]lim ⌊θ⌋ = 3 and [θ→3⁺]lim θ = 3, so this is just

3/3
1

For the second problem, note that if both limits exist and are finite (which can again be weakened to one being finite if suitable notions of addition and subtraction of infinity are defined), we have that [x→c⁺]lim (f(x)-g(x)) = ([x→c⁺]lim f(x)) - ([x→c⁺]lim g(x)). So again, just evaluate them one at a time:

[x→4⁺]lim x-⌊x⌋
([x→4⁺]lim x) - ([x→4⁺]lim ⌊x⌋)
4 - 4
0

I should point out that both of these problems were particularly easy because ⌊x⌋ is right-continuous, so evaluating its limit is as simple as just plugging in values. Had we been taking left-hand limits, we would have obtained:

[θ→3⁻]lim ⌊θ⌋/θ
([θ→3⁻]lim ⌊θ⌋) / ([θ→3⁻]lim θ)
2/3 (since [θ→3⁻]lim ⌊θ⌋ = 2, from the jump discontinuity)

[x→4⁻]lim x-⌊x⌋
([x→4⁻]lim x) - ([x→4⁻]lim ⌊x⌋)
4 - 3
1