When Limits Fail to Exist

Calculus 1 · Axiom Academy

LESSON When Limits Fail to Exist The three ways a two-sided limit can fail — a jump, a blow-up to infinity, and an endless oscillation — each shown as the behavior that breaks it. 1. What "the limit exists" requires A two-sided limit exists only when the left-hand limit and the right-hand limit agree on a single finite number . Break any part of that — agreement, finiteness, or settling at all — and the limit fails. There are exactly three ways it can break. The limit exists only when both one-sided limits hit the same finite L A jump discontinuity happens when both one-sided limits exist but disagree. Take the sign function at x = 0 : for any negative x it equals -1 , and for any positive x it equals +1 . The two sides head to different heights, so there is no single value to approach. Both one-sided limits exist — they just don't match Any step you take instantly — a price that jumps at a threshold, a piecewise rule that switches at a boundary — has this shape. The graph is two flat shelves with a gap between them. 3. The blow-up: unbounded near c An infinite discontinuity happens at a vertical asymptote : as the values grow without bound. For at x = 0 , the closer x gets to 0 — from either side — the larger f(x) becomes, racing up toward with no ceiling. 4. The oscillation: never settles

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