What is a Limit?

Business Calculus · Axiom Academy

The fundamental concept that makes calculus possible Before we can understand derivatives (rates of change) and integrals (accumulated quantities), we need one foundational concept: the limit . A limit describes what value a function approaches as the input gets closer and closer to some number—even if the function never actually reaches that value. Imagine you're driving toward a city. A limit is like asking: "What city are you heading toward?" rather than "Where are you right now?" You might be 10 miles away, then 5 miles, then 1 mile, then 100 feet... The limit is the destination you're approaching, even if you haven't arrived yet. Limits help us answer questions like: "As we produce more and more units, what does the average cost approach?" or "As time goes on, what will market share stabilize at?" Consider the function f(x) = 2x + 1 . What happens as x approaches 3? From both directions, f(x) is approaching 7 . We write: This is read: "The limit of 2x + 1 as x approaches 3 equals 7." Use the slider to see how f(x) = x^2 approaches 4 as x approaches 2: As x gets closer to 2 (from either side), f(x) gets closer to 4. The limit is 4, and in this case, f(2) actually equals 4 too! Limits become truly useful when we can't just plug in the value. Consider: What is f(2) ? If we try to plug in x = 2: Division by zero is undefined. We literally cannot compute f(2). But we can still ask: what value does f(x) approach as x gets close to 2?

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