Rational Function Limits Near a Cluster Point

This example uses an interactive GeoGebra applet to visualize how a rational function’s values behave in the neighborhood of a cluster point—either approaching a finite limit or revealing divergence/asymptotic behavior. The key mathematical ideas are limits, rational function behavior near critical points, and the geometric interpretation of continuity versus non-continuity using graph-level evidence. Construction is built with GeoGebra primitives such as Point, Segment, and Asymptote: key points and parameter positions (e.g., P, Q and their primed counterparts) are created and connected with segments to track approach paths, while Asymptote displays the relevant limiting boundary (e.g., vertical/horizontal/oblique behavior). The scene further uses labeled text objects to annotate the tracked values (such as y_P and y_Q) and organizes the workflow around the function and auxiliary objects to show the convergence pattern. Suitable for high-school advanced calculus learning and introductory university calculus, especially for building intuition about limits from graphs.