What Are Limits?

In calculus, limits help us understand what happens to a function as it approaches a specific point, even if it never actually gets there. Imagine zooming in closer and closer to a certain value on a graph—limits tell us the value the function is approaching.

Why Do Limits Matter?

Limits are the foundation of calculus. Without them, we wouldn’t be able to define derivatives or integrals. They allow us to describe instantaneous rates of change and the area under curves.

Key Ideas

• A limit tells us the value that a function gets close to as the input approaches a certain number.
• Not all limits exist—sometimes, the function behaves wildly as it nears a point.

How to Evaluate Limits

1. Direct Substitution: Plug the number into the function.
2. Simplify: Factor or cancel terms.
3. Use Special Limits: Know your basic limit rules, like the squeeze theorem.

Real-World Connection

Limits are used in engineering to predict system behaviors, in finance to model investment growth, and in science to describe rates of change.