To find the N-th Taylor polynomial for tan(x) at 0 that approximates the function within a specified error margin, the discussion emphasizes the need to express tan(x) as a power series derived from its known expansions for sin(x) and cos(x). The remainder term of the Taylor series is analyzed using the Lagrange form, which involves calculating higher-order derivatives of tan(x) to ensure the remainder is less than 10^-5. It is suggested that for N=5, the approximation will likely meet the error requirement based on the behavior of the series terms. The conversation also touches on the properties of alternating series and the conditions necessary for bounding the remainder. Ultimately, the participants conclude that evaluating the Taylor series up to a sufficient degree will yield the desired accuracy for approximating tan(x).