One-Sided Limits

Calculus Readiness · Axiom Academy

Approach from the left, approach from the right — and the two-sided limit lives only where those two agree. Take the step function f(x)=x+1 for x left , the outputs slide up toward 3 . Approaching from the right , they slide toward 1 . Same input a=2 , two different destinations — so we record two separate one-sided limits. Left-hand limit — approach through x < a Right-hand limit — approach through x > a 2. When Both Sides Agree, the Limit Exists Here the graph follows y=x+1 everywhere except it has a hole at x=1 . Coming from the left the outputs rise toward 2 ; coming from the right they fall toward 2 . Both tracers land on the same open point — so the ordinary two-sided limit exists and equals that shared value, 2 , regardless of the hole. — the outputs climb to 2 as x nears 1 from below. — the outputs drop to 2 as x nears 1 from above. Both equal 2 , so by the existence rule. Even if f(1) is undefined or set to some other height, the limit still reads 2 — limits ignore the point itself. The two-sided limit exists iff both one-sided limits exist and are equal. For , cancel to get x+1 for every . Both one-sided limits at 1 equal 2 , so — even though g(1) is the indeterminate and is undefined. 3. When the Sides Disagree, the Limit Fails

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