Motivated by the groundbreaking work of Andrews and Merca, truncated theta series have been extensively studied over the years. In particular, Merca made conjectures on the non-negativity of the coefficient of $q^N$ in truncated series from the Jacobi triple product identity and the quintuple product identity. In this paper, using Wright's Circle Method, we establish asymptotic formulas for the coefficients of truncated theta series and prove that Merca's conjectures are true for sufficiently large $N$.