limit as x approaches 0+ for x sin (1/x)

Question

lim x sin (1÷ x)

x --> 0⁺

Answer 1

lim_x→0 sin(1/x) d.n.e. because 1/x → ∞ and the lim_x→∞ sinx d.n.e. BUT ... sinx is bounded: it can only be a value between -1 and 1 inclusive. So the x in the product xsin(1/x) is pulling that product toward 0 and there is nothing (infinite) to stop it from doing that, so lim_x→0xsin(1/x) = 0. The right-handed limit doesn't matter, as the magnitude of the limit is 0 so the sign doesn't matter.

Graphing the function on desmos or another utility will show you this pretty cool function, which oscillates infinitely many times between x = 0 and x = 1, but the amplitude of those oscillations is tending toward 0, which is what makes the limit exist.