Prove limit as $x\to 0$ of $(1/x)\sin(1/x)$

Prove limit as $x\to 0$ of $(1/x)\sin(1/x)$

See my working below, hopefully you find it legible.Proving it using epsilon-delta, turned it into a proof by contradiction.I assume the limit converges to some L in the Reals, then consider two cases where L≥0 and where L<0 and want to show that in either case the limit does not converge to L.Therefore, the number L cannot exist and thus limit can't either.

Would appreciate some guidance on how to procede in that last step, not sure what 'epsilon' to pick to show that the expression in the absolute value is greater than epsilon.