This presentation highlights the objectives of a study focusing on two areas of research. Firstly, it investigates the impact of oblique asymptotes on the graphical behavior of rational functions. By comprehending the fundamental principles of rational functions and their asymptotic behavior, the study establishes the conditions for a rational function to possess an oblique asymptote. Through in-depth analysis using calculus, algebraic manipulation, and graphical representation, the behavior of rational functions in proximity to oblique asymptotes is examined. The exploration of various scenarios involving intersections or coincidences with other asymptotes sheds light on the interplay between different asymptotic behaviors, particularly for quadratic by linear rational functions. Secondly, the study addresses the challenges faced by students when solving first-order differential equations using four different methods. By scrutinizing student answer scripts and identifying various problem-solving errors, the research aims to enhance teaching strategies and emphasize crucial points in the problem-solving process.