limits -- Prove that xsin1/x approaches 0 near 0 Prove that xsin1/x approaches 0 near 0. Similiar Proof from book |sin1/x| ≤ 1 | xsin1/x | ≤ |x| for all x not equal to 0, so we can make |xsin1/x|< ε by requiring that |x| < ε and not equal to 0. MY QUESTION: Prove x2sin1/x approaches 0 near 0. According to the book, if ε>0, to ensure that |x2sin1/x |< ε we need only require that |x| < ε and not equal to 0. shouldnt |x| be less that √ε? Hence |x2 | < ε and |x|< √ε.