How do I graph this "Greatest Integer Function" ?

Question

it looks like this:

lim L x ||
x => 3+

the brackets around the x are wierd looking, like to two capital L's. I have no idea how to graph it... It's a limit problem, so I have to find the limit for it, as x approaches 3 from the left.

Answer 1

Limit from the left or from the right? I ask because left-hand limits are denoted using a minus, not a plus.

Anyway, the floor function is defined to be the largest integer not greater than x. Basically, we take the input value x, and we push it down until we reach an integer - the first integer we hit is ⌊x⌋. This should be easier for you to visualize than the more abstract definition. Another way to understand the floor function is to consider the behavior of this function over a smaller range, say, [n, n+1), where n is some integer. Clearly, n is less than or equal to every number in that range, and if k is any integer larger than n, then k≥n+1>x, so k is not less than or equal to x. This means that the largest integer less than or equal to x is in fact n itself -- that is, x∈[n, n+1) ⇒ ⌊x⌋=n. And this is true for every n, so we can write the floor function as a piecewise function as follows:

⌊x⌋={n if n≤x
This gives you an immediate idea of what the graph of the function looks like - for each integer n, draw a horizontal line from (n, n) to (n, n+1), put a closed dot at the left end and an open dot at the right end. It should look like a staircase (see http://en.wikipedia.org/wiki/Floor_function).

This also should tell you how to find the limit -- since the function is piecewise constant, simply treat it as you would any other piecewise function. In this case, 3 is on the boundary of two pieces, namely [2, 3) and [3, 4). On [2, 3) ⌊x⌋=2 so we have that:

[x→3⁻]lim ⌊x⌋ = [x→3⁻]lim 2 = 2

and on [3, 4) we have that ⌊x⌋=3, so:

[x→3⁺]lim ⌊x⌋ = [x→3⁺]lim 3 = 3

These limits are not equal, so the function is not continuous at x=3. The floor function has the same sort of jump discontinuity at every integer.

I hope this was helpful to you.

Answer 2

this is how the graph looks like...
http://www.pen.k12.va.us/Div/Winchester/jhhs/math/gifs2/grtint.jpg

as you can see, as x approaches 3 from the left, the limit is 2..

if x approaches 3 from the right (which is what x->3+ means), the limit is 3..