Introduction to Limits

AP Calculus · Axiom Academy

What happens when we get really, really close? Imagine a car driving down a road. At any given moment, its speedometer shows a speed—say, 60 mph. But how does the car actually "have" that speed? It's not the distance traveled in an hour (that would be confusing because it's only been driving for a few seconds). Instead, it's measuring something instantaneous: how fast is it going right now ? The answer is limits. Instead of looking at what happens at a point, we look at what happens near that point. We ask: "What value is the function approaching as the input gets closer and closer to our point?" A ball falls from a cliff. Its height at time t is given by . If we want to know the instantaneous velocity at t = 2 seconds, we look at the average velocity over smaller and smaller time intervals: From t=2 to t=2.1: average velocity = ___ From t=2 to t=2.01: average velocity = ___ From t=2 to t=2.001: average velocity = ___ (getting closer to the limit!) What is the instantaneous rate of return on an investment? As the time interval shrinks, we approach the true instantaneous rate. How fast is a tumor growing right now? We look at how much it grows in smaller and smaller time periods. What is the instantaneous acceleration? The rate at which velocity is changing at this very moment? What You'll Learn in This Unit ✓ Graphical interpretation: What does a limit look like on a graph? ✓ Numerical approach: How do we calculate limits by plugging in numbers?

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