Finding Limits from Graphs

Calculus Readiness · Axiom Academy

LESSON Reading Limits from Graphs Before any algebra, a limit is something you can see : walk the curve toward a point from both sides and watch where the height is heading. 1. Trace Both Sides Toward a Point To read , put one finger on the curve to the left of x = a and slide it toward a ; do the same from the right . Watch the height each finger is heading for. In the animation, two tracer dots ride the curve toward x = 2 and drop their heights onto the y -axis. They meet at the same place — so that height is the limit. Both one-sided limits agree on the same height L so the two-sided limit equals that height 2. A Hole Doesn't Break the Limit Now the curve has a hole at x = 1 : an open circle where the point is missing. The two tracers still glide toward that spot from both sides and still agree on a height — the open circle at y = 2 . The function's posted value is somewhere else entirely: a solid dot pasted on at y = 4 . The limit cares about the approach, not the single posted point. Sliding in from x < 1 , the height heads to 2 : . Sliding in from x > 1 , the height also heads to 2 : . Both sides agree, so — the height of the open circle. The solid dot says f(1) = 4 . An isolated exception cannot change the approach. Limit and value are separate questions

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