Graphical Understanding of Limits

AP Calculus · Axiom Academy

LESSON Graphical Understanding of Limits Visualize how functions approach values as x approaches specific points A limit describes the value that a function approaches as the input approaches some value. The key insight is that we care about how the function behaves near a point , not necessarily at that point itself. We say the limit of as is if we can make as close to as we want by taking sufficiently close to . When you're given a graph, finding limits is visual: Find the x-value where you want the limit Trace upward to see the function behavior approaching from the left Trace upward from the right side The limit exists if both sides approach the same y-value might not equal . The function might not even be defined at ! This has a hole at x = 2, but because it approaches 4 from both sides. One-Sided vs. Two-Sided Limits A two-sided limit exists only if both one-sided limits exist and are equal:

This is the written version of the interactive lesson above. See the full AP Calculus course.