question about greatest integer function with limits?

Question

how would i go about solving this problem?...

If [x] is the greatest integer function, find

the limit as x goes to 2 from the left of

(x - [x]) / [x-3]

Answer 1

Solving this step by step we get

lim{x --> 2-} (x - [x]) / [x - 3]
= (lim{x --> 2-} (x - [x])) / (lim{x --> 2-} [x - 3]) = N / D

This holds since (we assume for now that) both numenator and denominator are defined.

N = (lim{x --> 2-} x) - lim{x --> 2-} [x] = 2 - 1 = 1

because [x] = 1 for all 1 < x < 2.

Analogously, we find that

D = lim{x --> 2-} [x - 3] = -2

because [x - 3] = -2 for all 1 < x < 2.

Hence the limit of f(x) as x converges to 2 from the left equals N/D = -1/2.

Answer 2

Since at x = 2 the denominator doesn't have a zero in it, there's no problem with evaluating the function directly at that point. So at x = 2
(x - [x])/(x-3) = (2 - 2)/(2 - 3) = 0/-1 = 0


Doug

Answer 3

the function is discontinuous at all integer values of x
0<=(x-[x])<1

x f(x)
0.0 0.0
0.5 , 0.5/-3
1.0 , 0.0
1.5 , 0.5/-2=-0.25
1.9 , 0.9/-2=-0.45
1.99 , 0.99/-2=0.495

Intuitively (or by induction, if you prefer), the limit is 0.5, if it can be said to exist at all.