Evaluating Limits Graphically

Pre-Calculus · Axiom Academy

LESSON Evaluating Limits Graphically Read a limit straight off a graph by watching where the curve is headed from each side. 1. The Limit Is Where the Curve Is Headed To read from a graph, walk along the curve toward x = c — once from the left and once from the right — and watch the height the curve aims for. Here f(x) = x , and as from both sides the height homes in on 2 . Both sides aim for the same height — that height is the limit 2. A Hole Does Not Stop the Limit A hole (removable discontinuity) is a single point where the function is undefined, drawn as an open circle. The two-sided walk still works: the curve approaches the same height from both sides even though it never lands there. Here the curve sits at height 3 everywhere except the hole at x = 2 . Even though f(2) is undefined (the open circle), the limit is 3 because both sides head to 3 . 3. A Jump Splits the Two One-Sided Limits At a jump , the left and right sides head to different heights, so the two-sided limit does not exist. We can still name each side separately with a one-sided limit : a for the approach from the left, a for the approach from the right. The one-sided limits disagree ( ), so does not exist. 4. A Vertical Asymptote: the Curve Runs Off the Page When the curve shoots toward or as , we have a vertical asymptote at x = c . For the left side plunges to and the right side rockets to . 5. Far Out on the Axis: Limits at Infinity

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