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

Key Ideas

1. Limits describe behavior near a point, not necessarily at the point
2. The function doesn't need to be defined at $a$ for the limit to exist
3. We care about what happens as we get arbitrarily close to $a$ from both sides

A Simple Example

Consider $f(x) = 2x + 1$. What happens as $x$ approaches 3?

$\lim_{x \to 3} (2x + 1) = 2(3) + 1 = 7$

As $x$ gets closer to 3, $f(x)$ gets closer to 7.

Why Limits Matter

Limits allow us to:

• Define derivatives (rates of change)
• Define integrals (accumulated change)
• Handle discontinuities
• Work with infinity
• Make calculus rigorous and precise