What is a Limit?

Understanding Limits Intuitively

A limit describes what happens to a function as the input gets closer and closer to a certain value. Think of it as asking: "Where is this function heading?"

Key Idea: Limits are about the journey, not the destination. We care about where a function is going, not necessarily where it is.

The Limit Notation

We write:

$\lim_{x \to a} f(x) = L$

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

What This Means

• $x \to a$: The variable x gets closer and closer to the value a
• $f(x) \to L$: The function values get closer and closer to L
• We don't care about what happens at x = a, only what happens near it

A Simple Example

Consider the function: $f(x) = 2x + 1$

What is $\lim_{x \to 3} (2x + 1)$?

As x gets close to 3:

• When x = 2.9: $f(2.9) = 2(2.9) + 1 = 6.8$
• When x = 2.99: $f(2.99) = 2(2.99) + 1 = 6.98$
• When x = 2.999: $f(2.999) = 2(2.999) + 1 = 6.998$

The values are getting closer to 7!

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

Why Limits Matter

Limits are the foundation of calculus because they help us:

• Define derivatives (rate of change)
• Define integrals (area under curves)
• Understand function behavior near problematic points
• Handle infinity and infinitesimal quantities

Important Note

The limit of f(x) as x approaches a might be different from f(a), or f(a) might not even exist!