This paper presents a numerical investigation of singularly perturbed boundary value problems (SPBVPs) using the spectral method. These types of differential equations arise frequently in physics and engineering, where the presence of a small parameter leads to sharp gradients or boundary layers. Standard numerical methods often fail to capture these features efficiently. In this study, we apply Chebyshev spectral collocation techniques to approximate the solution of a singularly perturbed second-order differential equation. We analyze the accuracy, convergence, and stability of the method, and compare the results with exact or asymptotic solutions. Numerical experiments confirm that the spectral method provides highly accurate approximations even in the presence of strong boundary layers.