What is a Limit? (Intuitive Approach)

Calculus Readiness · Axiom Academy

The single idea calculus is built on: not what a function equals at a point, but what it approaches . 1. The Idea: Closing In on a Value Take the friendly function f(x) = x + 2 and ask what happens as x approaches 3 . We are not asking for f(3) — we are asking what f(x) gets close to . Watch two points crawl toward x=3 , one from the left and one from the right. Their heights are pulled toward the same value. Read it: "the limit of f(x) , as x approaches 3 , equals 5 ." 2. The Payoff: A Limit at a Hole Now the interesting case. Consider . Plug in x = 1 and you get — undefined. The function has a literal hole there. Yet as x approaches 1 from either side, the output marches straight toward 2 . The moving point in the animation reaches the gap, the hole stays empty, and the value sails through to 2 anyway. Factor the top: x^2 - 1 = (x-1)(x+1) . As long as the (x-1) cancels, leaving g(x) = x + 1 . As , that approaches 2 : g(1) is undefined , but . A limit does not care what happens at x = a — only what the function is doing near it. 3. The Point at a Is Irrelevant Here is the heart of it. The limit is set entirely by the curve's approach from both sides — so we can move, or even delete, the single value at x = a and the limit does not flinch. Watch the dot at a cycle through three lives — sitting on the curve, displaced to a wrong height, then gone — while the approaching arrows keep pointing at the same L .

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