Summary: In a Hilbert space \(\mathfrak{H} \), a family of operators \(A(t)\), \(t\in \mathbb{R} \), is treated admitting a factorization of the form \(A(t) = X(t)^* X(t)\), where \(X(t)=X_0+X_1t+\cdots +X_pt^p\), \(p\ge 2\). It is assumed that the point \(\lambda_0=0\) is an isolated eigenvalue of finite multiplicity for \(A(0)\). Let \(F(t)\) be the spectral projection of \(A(t)\) for the interval \([0,\delta ]\). For \(|t| \le t^0\), approximation in the operator norm in \(\mathfrak{H}\) for the projection \(F(t)\) with an error \(O(t^{2p})\) is obtained as well as approximation for the operator \(A(t)F(t)\) with an error \(O(t^{4p})\) (the so-called threshold approximations). The parameters \(\delta\) and \(t^0\) are controlled explicitly. Using threshold approximations, approximation in the operator norm in \(\mathfrak{H}\) is found for the resolvent \((A(t)+\varepsilon^{2p}I)^{-1}\) for \(|t|\le t^0\) and small \(\varepsilon >0\) with an error \(O(1)\). All approximations mentioned above are given in terms of the spectral characteristics of the operator \(A(t)\) near the bottom of the spectrum. The results are aimed at application to homogenization problems for periodic differential operators in the small period limit.