Babynini
  • Babynini
greatest integer functions. Help!
Mathematics
  • Stacey Warren - Expert brainly.com
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chestercat
  • chestercat
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Babynini
  • Babynini
#51
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Babynini
  • Babynini
@SolomonZelman you free? :)
jim_thompson5910
  • jim_thompson5910
do you know how the greatest integer function is defined?

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Babynini
  • Babynini
Mm I know what the graph of it looks like. But not quite sure how it's defined
jim_thompson5910
  • 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
  • jim_thompson5910
greatest integer function = floor function
jim_thompson5910
  • jim_thompson5910
think of the number line written vertically |dw:1444263692393:dw|
jim_thompson5910
  • 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
  • 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
  • Babynini
approaching it from either the positive or negative side we always do that?
SolomonZelman
  • 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
  • jim_thompson5910
let's try an example if f(x) = [[x]], then what is f(8.3125) equal to?
Babynini
  • Babynini
8?
jim_thompson5910
  • jim_thompson5910
yes, how about f(8.99999)
Babynini
  • Babynini
8 still
jim_thompson5910
  • jim_thompson5910
good, so if the number is positive, you just chop off the decimal portion
jim_thompson5910
  • jim_thompson5910
now let's try f(-2.462)
Babynini
  • Babynini
Would that round up? to 3?
jim_thompson5910
  • jim_thompson5910
I think you meant -3
jim_thompson5910
  • jim_thompson5910
look back to the vertical number line
Babynini
  • Babynini
yes yes sorry o.o
jim_thompson5910
  • jim_thompson5910
|dw:1444264095476:dw|
jim_thompson5910
  • 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
  • Babynini
so approaching -2 from the right = -2 and from the left = -2 ?
Babynini
  • Babynini
because there's no decimals o.o
jim_thompson5910
  • jim_thompson5910
look at the rules SolomonZelman posted
Babynini
  • Babynini
Ahh ok. Approaching - 2 from the right = 2 approaching - 2 from the left = ...1?
jim_thompson5910
  • jim_thompson5910
idk how you jumped from -2 to +2
Babynini
  • Babynini
*-2 and -3
jim_thompson5910
  • jim_thompson5910
\[\Large \lim_{x \to -2^{+}} [[x]] = -2\] \[\Large \lim_{x \to -2^{-}} [[x]] = -3\] looks good
Babynini
  • Babynini
Sorry, my brains are not working haha =.=
jim_thompson5910
  • jim_thompson5910
does \[\Large \lim_{x \to -2^{}} [[x]]\] exist?
Babynini
  • Babynini
and then as x approaches -2.4 it = -3
Babynini
  • Babynini
No, it doesn't.
jim_thompson5910
  • jim_thompson5910
why not?
Babynini
  • Babynini
because the points aren't meeting
jim_thompson5910
  • jim_thompson5910
yes, specifically because \(\Large \text{LHL} \ne \text{RHL}\) LHL = left hand limit RHL = right hand limit
jim_thompson5910
  • jim_thompson5910
so 51(a)(ii) does not exist
Babynini
  • Babynini
ahh gotcha. Okay!
Babynini
  • Babynini
Part b!
Babynini
  • Babynini
for c) the answer is: for all non-integer values of a, yeah?
jim_thompson5910
  • jim_thompson5910
`for c) the answer is: for all non-integer values of a, yeah?` agreed
jim_thompson5910
  • jim_thompson5910
part b) is very similar to what SolomonZelman wrote out
Babynini
  • Babynini
Okay, I wasn't sure if that was too "simple" So b: i) n-1 ii) n
jim_thompson5910
  • jim_thompson5910
looks good
Babynini
  • Babynini
Thank you so much :)
jim_thompson5910
  • jim_thompson5910
sure thing
Empty
  • 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
  • Babynini
hahah my next problem is that! i'll tag you in it ;)
Babynini
  • Babynini
Though from all you said here I think I could get it.

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