Expert: Alon Mandes - 2/12/2009

Question
For each of the following fuinctions from S to R (with f(x) being set to 0 whenever the definition for
f(x) does not make sense). state (with explanation):

(i) whether it is continous on its domain S;
(ii) if not, where the points of discontinuity are and of what type are these discontinuites (removable.
jump, or infinite): and
(iii) the equations of any horizontal or vertical asymptotes.
1) f(x)= (x sin (1/x)) - 2x ; S=( -infinite, + infinite)
2) f(x)= x/x^2+1   ; S=(-infinite, +infintite)
3) f(x) = (1/rootsinx)-1   ; S= [0, pi]
4) f(x) = arctan(x-0.5/x^2-0.1x-0.7)   ; S=[-1.5,1.5]
5) f(x) = x - [x]  ; S=(-infinite, +infinite)

Answer
f:S->R


1)f(x)= (x sin (1/x)) - 2x ; S=( -infinite, + infinite)
f is not continous on its domain S; x=0 is jump point.no asymptots.

5) f(x) = x - [x]  ; S=(-infinite, +infinite)
f is not continous on its domain S, it's a jump point;no asymptots.

3) f(x) = (1/rootsinx)-1   ; S= [0, pi]
f is continous on its domain S. no asymptots.

2) f(x)= x/x^2+1   ; S=(-infinite, +infintite)
f is continous on its domain S; y=0 is the horizontal asymptot.

4) f(x) = arctan(x-0.5/x^2-0.1x-0.7)   ; S=[-1.5,1.5]
f is continous on its domain S. no asymptots.