Functions and Limits

TL;DR

Functions describe how one quantity depends on another, letting us model real-world relationships. Limits help us understand a function's behavior as its input gets incredibly close to a specific value, even if the function isn't defined at that exact point. Together, they form the bedrock of calculus, allowing us to study change and continuity.

1. The Mental Model

Think of a function as a machine: you put something in (an input), and it consistently spits something out (an output) according to a rule. A limit is like trying to predict exactly where a moving object will be when it reaches a certain point, even if it briefly disappears or teleports at that point.

2. The Core Material

What's a Function?

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A function assigns exactly one output for every input. If you put in 'x', you always get the same 'y' out. We write it as f(x). For example, f(x) = x + 2 means if you input x=3, you output y=5. You can't put in x=3 and sometimes get 5 and sometimes 7.

Understanding Limits

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A limit tells us what value a function is approaching as its input approaches some number. It's about the function's behavior near a point, not necessarily at the point itself. We write this as:

lim (x→a) f(x) = L

This reads: "The limit of f(x) as x approaches a is L."

Why do we need limits?

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