Foundations of Limits and Continuity

TL;DR

Limits describe the value a function approaches as its input gets arbitrarily close to a certain point, even if the function isn't defined there. Continuity means a function's graph has no breaks, jumps, or holes, which we can formally check using limits. Understanding these concepts is fundamental for calculus since derivatives and integrals are built upon them.

1. The Mental Model

Imagine walking along a path (your function's graph). A limit tells you where you'd be headed at a specific spot. If the path is continuous, you can walk through that spot without lifting your feet.

2. The Core Material

What is a Limit?

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A limit is about the "tendency" of a function. When we write $\lim_{x \to a} f(x) = L$, it means that as $x$ gets closer and closer to $a$ (from both sides, but not necessarily equal to $a$), the value of $f(x)$ gets closer and closer to $L$.

Think of it like this: if you plug in numbers slightly less than $a$ and numbers slightly greater than $a$ into $f(x)$, the outputs should get very close to $L$.

It doesn't matter what $f(a)$ actually is, or even if $f(a)$ exists. The limit is purely about the function's behavior near $a$.

One-Sided Limits

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Sometimes, a function behaves differently depending on which side you approach $a$ from.
* Left-hand limit: $\lim_{x \to a^-} f(x) = L_1$ means $f(x)$ approaches $L_1$ as $x$ approaches $a$ from values less than $a$.
* Right-hand limit: $\lim_{x \to a^+} f(x) = L_2$ means $f(x)$ approaches $L_2$ as $x$ approaches $a$ from values greater than $a$.

For the overall limit $\lim_{x \to a} f(x)$ to exist, the l