## Babynini one year ago greatest integer functions. Help!

1. Babynini

#51

2. Babynini

@SolomonZelman you free? :)

3. jim_thompson5910

do you know how the greatest integer function is defined?

4. Babynini

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

5. 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

6. jim_thompson5910

greatest integer function = floor function

7. jim_thompson5910

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

8. 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|

9. 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|

10. Babynini

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

11. 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.

12. jim_thompson5910

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

13. Babynini

8?

14. jim_thompson5910

15. Babynini

8 still

16. jim_thompson5910

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

17. jim_thompson5910

now let's try f(-2.462)

18. Babynini

Would that round up? to 3?

19. jim_thompson5910

I think you meant -3

20. jim_thompson5910

look back to the vertical number line

21. Babynini

yes yes sorry o.o

22. jim_thompson5910

|dw:1444264095476:dw|

23. 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

24. Babynini

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

25. Babynini

because there's no decimals o.o

26. jim_thompson5910

look at the rules SolomonZelman posted

27. Babynini

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

28. jim_thompson5910

idk how you jumped from -2 to +2

29. Babynini

*-2 and -3

30. jim_thompson5910

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

31. Babynini

Sorry, my brains are not working haha =.=

32. jim_thompson5910

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

33. Babynini

and then as x approaches -2.4 it = -3

34. Babynini

No, it doesn't.

35. jim_thompson5910

why not?

36. Babynini

because the points aren't meeting

37. jim_thompson5910

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

38. jim_thompson5910

so 51(a)(ii) does not exist

39. Babynini

ahh gotcha. Okay!

40. Babynini

Part b!

41. Babynini

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

42. jim_thompson5910

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

43. jim_thompson5910

part b) is very similar to what SolomonZelman wrote out

44. Babynini

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

45. jim_thompson5910

looks good

46. Babynini

Thank you so much :)

47. jim_thompson5910

sure thing

48. 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.

49. Babynini

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

50. Babynini

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