greatest integer functions. Help!

- Babynini

greatest integer functions. Help!

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- Babynini

#51

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- Babynini

@SolomonZelman you free? :)

- jim_thompson5910

do you know how the greatest integer function is defined?

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## More answers

- Babynini

Mm I know what the graph of it looks like. But not quite sure how it's defined

- jim_thompson5910

http://mathbits.com/MathBits/TISection/PreCalculus/GraphGreatestIntFunction.html
the basic idea is that the input x could be any real number
the output is always a whole number (positive or negative). You round down to the nearest whole number

- jim_thompson5910

greatest integer function = floor function

- jim_thompson5910

think of the number line written vertically
|dw:1444263692393:dw|

- jim_thompson5910

if we plug in x = 1.7 then [[x]] = [[1.7]] = 1
we round down to the nearest whole number which in this case is 1
|dw:1444263763896:dw|

- jim_thompson5910

now let's say we plugged in x = -0.16
that would move to -1 because we're moving down the ladder or building
|dw:1444263816583:dw|

- Babynini

approaching it from either the positive or negative side we always do that?

- SolomonZelman

Here are the properties, which follow just from logic:
*[1]* \(\Large\color{black}{ \displaystyle \lim_{x~\to~a^+} \left[\left[x\right]\right]=a }\)
*[2]* \(\Large\color{black}{ \displaystyle \lim_{x~\to~a^-} \left[\left[x\right]\right]=a-1 }\)
where *a* is an integer.

- jim_thompson5910

let's try an example
if f(x) = [[x]], then what is f(8.3125) equal to?

- Babynini

8?

- jim_thompson5910

yes, how about f(8.99999)

- Babynini

8 still

- jim_thompson5910

good, so if the number is positive, you just chop off the decimal portion

- jim_thompson5910

now let's try f(-2.462)

- Babynini

Would that round up? to 3?

- jim_thompson5910

I think you meant -3

- jim_thompson5910

look back to the vertical number line

- Babynini

yes yes sorry o.o

- jim_thompson5910

|dw:1444264095476:dw|

- jim_thompson5910

think of each whole number as a floor in a building
-2.462 is between the two floors
the floor function moves -2.462 down to the nearest floor

- Babynini

so approaching -2
from the right = -2
and from the left = -2 ?

- Babynini

because there's no decimals o.o

- jim_thompson5910

look at the rules SolomonZelman posted

- Babynini

Ahh ok.
Approaching - 2 from the right = 2
approaching - 2 from the left = ...1?

- jim_thompson5910

idk how you jumped from -2 to +2

- Babynini

*-2
and -3

- jim_thompson5910

\[\Large \lim_{x \to -2^{+}} [[x]] = -2\]
\[\Large \lim_{x \to -2^{-}} [[x]] = -3\]
looks good

- Babynini

Sorry, my brains are not working haha =.=

- jim_thompson5910

does \[\Large \lim_{x \to -2^{}} [[x]]\] exist?

- Babynini

and then as x approaches -2.4 it = -3

- Babynini

No, it doesn't.

- jim_thompson5910

why not?

- Babynini

because the points aren't meeting

- jim_thompson5910

yes, specifically because \(\Large \text{LHL} \ne \text{RHL}\)
LHL = left hand limit
RHL = right hand limit

- jim_thompson5910

so 51(a)(ii) does not exist

- Babynini

ahh gotcha. Okay!

- Babynini

Part b!

- Babynini

for c) the answer is: for all non-integer values of a, yeah?

- jim_thompson5910

`for c) the answer is: for all non-integer values of a, yeah?`
agreed

- jim_thompson5910

part b) is very similar to what SolomonZelman wrote out

- Babynini

Okay, I wasn't sure if that was too "simple"
So b:
i) n-1
ii) n

- jim_thompson5910

looks good

- Babynini

Thank you so much :)

- jim_thompson5910

sure thing

- Empty

Oh I thought this was going to be another squeeze theorem thing, but in case you come across one soon, I think you might be entertained:
\[x \le [[x]] \le x+1\]
So if you're given like find the limit of \(x [[x]]\) as x approaches 0 you already know:
\[x^2 \le x[[x]] \le x^2+x\]
So
\[\lim_{x \to 0} x[[x]] = 0\]
Anyways there are a lot of problems with squeeze theorem and this rounding function in it so I thought it'd be fun to share real fast for fun.

- Babynini

hahah my next problem is that! i'll tag you in it ;)

- Babynini

Though from all you said here I think I could get it.

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