For various but sound practical reasons it has become desirable to approximate to previously tabulated mathematical functions by polynomials or rational fractions. In this paper Chebyshev polynomials are used to approximate Mills' Ratio over two separate ranges [0, 1], [1, $\infty$] of the argument. Some new asymptotic expansions for this ratio are also obtained by an extended use of the symbolic operator method, revealing incidentally that Ruben's (1962) expansion is a special but not necessarily superior case.