Limits

A limit describes what value a function is approaching as the input gets close to a certain point. The key idea is that we care about nearby behavior, not necessarily what happens at the point itself. Instead of asking “What is the value here?”, limits ask “What happens as we get closer?” By examining values from both sides and looking at trends, we can predict behavior even when a function is undefined or behaves strangely at a point.

A one-sided limit describes how a function behaves as the input approaches a point from only one direction—either from the left or from the right. This section focuses on observing directional behavior using graphs and values, and on understanding why approaching from different sides can lead to different outcomes. One-sided limits help explain jumps, boundaries, and points where a function changes behavior.

An infinite limit describes a situation where a function grows without bound as the input approaches a certain value. Instead of approaching a finite number, the function’s values increase or decrease indefinitely. In this section, you will explore how graphs behave near vertical asymptotes and learn to recognize unbounded behavior by observing patterns rather than memorizing rules.

Algebraic techniques are methods for rewriting expressions so limit behavior becomes clearer. The goal is not to “plug in and calculate,” but to remove obstacles that hide what a function is approaching. In this section, you will experiment with factoring, simplifying, and rearranging expressions to reveal hidden structure and make limits easier to understand.

A function is continuous at a point if its value matches what the function is approaching at that point. Informally, the graph can be drawn without lifting your pencil. This section explores continuity by testing conditions, examining graphs, and identifying breaks such as holes, jumps, and asymptotes. The emphasis is on recognizing why continuity fails, not just labeling it.