Finding Limits from Tables

Calculus Readiness · Axiom Academy

LESSON Finding Limits from Tables Estimate a limit numerically: plug in values that creep toward a from both sides and watch the outputs home in. 1. The Idea: Squeeze In and Watch If you can't easily evaluate f at x=a , evaluate it near a instead. Pick x -values like creeping in from the left, and creeping in from the right. If the outputs settle on a single number L from both directions, that L is your estimate of the limit. Pick a few x -values approaching a from the left . Pick a few x -values approaching a from the right . Look at what number the outputs are closing in on. Watch it happen for the famous limit (with x in radians). Plugging in x=0 gives , and there is no obvious factoring — so a table is the natural tool. The rows fill in from the outside in; keep your eye on the right-hand column. Here is the same table written out, with one x -value per column instead of per row: Estimate . Plugging in x=2 gives again — the function has a hole there — so we build a table marching toward 2 from both sides. As we squeeze in on x=2 from both sides, the outputs close in on 4 . for , which approaches 2+2=4 . The table got it exactly right — tables and algebra agree. A table doesn't only confirm limits — it also warns you when one doesn't exist. There are two telltale patterns, and the animation shows them filling in side by side. Watch the output columns: one set never settles, the other settles on two different values.

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