Give it the squeeze.
-1 ≤ sin(1/x) ≤ 1 for all x ≠ 0
-x ≤ xsin(1/x) ≤ x
As x approaches zero,
lim -x = 0 and lim x = 0.
Therefore, lim xsin(1/x) = 0.
-1 ≤ sin(1/x) ≤ 1 for all x ≠ 0
-x ≤ xsin(1/x) ≤ x
As x approaches zero,
lim -x = 0 and lim x = 0.
Therefore, lim xsin(1/x) = 0.