Evaluating Limits Graphically

Calculus 1 · Axiom Academy

EXAMPLE Evaluating Limits Graphically Read one-sided limits, the two-sided limit, and the function value straight off a graph. The graph below shows a piecewise function f . Using the graph, evaluate the left-hand limit, the right-hand limit, and the two-sided limit of f as , then compare them with the actual function value f(1) . Nice work. You read every limit at x = 1 off the graph and saw why the limit can differ from the function value. One-sided limits: trace the curve toward x = 1 from each side. Here both sides head to 2 , so . Two-sided limit: it exists only when the two sides agree. They do, so . Open vs. closed circle: the open circle at (1, 2) is approached but not attained; the closed circle gives the real value f(1) = 3 . Limit value: while f(1) = 3 . The function has a hole at x = 1 — a removable discontinuity. A limit describes where a function is heading ; the function value is where it actually is . They usually match — but a graph makes it plain when they don't.

This is the written version of the interactive lesson above. See the full Calculus 1 course.