The Concept of a Limit

Calculus 1 · Axiom Academy

What it means for a function to approach a value — and why what happens AT a point doesn't decide the limit. 1. A Limit Is Where a Function Is Headed A limit describes what value a function approaches as the input gets closer and closer to some number. It's not about what happens AT that point — it's about where the function is headed as you close in on it. means f(x) gets arbitrarily close to L as x approaches c — without necessarily equalling L when x = c . Watch f(x) = x^2 + 1 as x closes in on c = 2 from both sides . The input slides toward 2 ; the output is squeezed toward the same height — the limit L = 5 . c — the value the input x is approaching L — the value the output f(x) is approaching — "approaches" / "gets closer and closer to" 2. Left and Right: One-Sided Limits Sometimes a function behaves differently depending on which way you come in. You can approach c from two directions , and each gives its own one-sided limit. Approach from below, . Written — the minus means "from the left." Approach from above, . Written — the plus means "from the right." Here the left side climbs toward 3 while the right side climbs toward 5 . Watch the two tracers arrive at different heights — that gap is the whole point of one-sided limits. The open circles sit at (2,3) and (2,5) : neither is "the" value — and the function value right at x = 2 might be a third number, or undefined entirely. The left and right limits simply record where each side is headed.

This is the written version of the interactive lesson above. See the full Calculus 1 course.