What are the asymptote(s) and hole(s), if any, of # f(x) = xsin(1/x)#?
2 Answers
Refer below.
Explanation:
Well, there is obviously a hole at
We can graph the function:
graph{xsin(1/x) [10, 10, 5, 5]}
There are no other asymptotes or holes.
It also has a horizontal asymptote
It has no vertical or slant asymptotes.
Explanation:
Given:
#f(x) = x sin(1/x)#
I will use a few of properties of

#abs(sin t) <= 1" "# for all real values of#t# . 
#lim_(t>0) sin(t)/t = 1# 
#sin(t) = sin(t)" "# for all values of#t# .
First note that
#f(x) = (x) sin(1/(x)) = (x)(sin(1/x)) = x sin(1/x) = f(x)#
We find:
#abs(x sin(1/x)) = abs(x) abs(sin (1/x)) <= abs(x)#
So:
#0 <= lim_(x>0+) abs(x sin(1/x)) <= lim_(x>0+) abs(x) = 0#
Since this is
Also, since
#lim_(x>0^) x sin(1/x) = lim_(x>0^+) x sin(1/x) = 0#
Note that
We also find:
#lim_(x>oo) x sin(1/x) = lim_(t>0^+) sin(t)/t = 1#
Similarly:
#lim_(x>oo) x sin(1/x) = lim_(t>0^) sin(t)/t = 1#
So
graph{x sin(1/x) [2.5, 2.5, 1.25, 1.25]}