Polynomial tails prevent exponential tilting

If a density has polynomially decaying tails on both half-lines, its moment-generating function is infinite at every nonzero real parameter: exponential growth defeats polynomial decay on the positive tail for a positive parameter, and on the negative tail for a negative parameter. Thus no open real tilting domain exists for a saddlepoint density approximation, even when the distribution has finite mean and variance and a central limit theorem applies.