In the theory of polynomials orthogonal with respect to an inner product of the form f, = ∫^∞₀f(x)g(x)dψ(x)+Σ^m_k=1A_kf^(i_k)(0)g^(i_k)(0), one is confronted with the following situation: for certain values of the parameters, the orthogonal polynomial of degree n does not have all its zeros inside the support of the distribution function dψ. This paper gives a method to investigate the zero distribution by looking at a type of limiting polynomial. For the case m = 2 it is shown that there are exactly two zeros outside the true interval of orthogonality for A₁,A₂ large; moreover, it is proved that these zeros are nonreal (complex conjugates) in the case i₁ + 1 = i₂. Also several examples are given.