1. The Reality Check

Limits are the foundation calculus is built on — they let mathematicians rigorously handle infinity and division by zero, situations that break ordinary algebra.

2. Plain English Definition

A limit describes what value a function approaches as the input gets infinitely close to a specific point, even if the function is never actually defined there.

3. When Direct Substitution Fails

Usually you can just plug in a number. But expressions like (x²−1)/(x−1) as x approaches 1 give 0/0 — an "indeterminate form" that requires more work, not a dead end.

4. Key Terms

• Indeterminate Form: An expression like 0/0 that requires further simplification to resolve.
• Continuity: A function is continuous where its limit equals its actual value at that point.

Frequently Asked Questions

Does a limit always equal the function's actual value there?

Not necessarily — a function can have a "hole" at a point while its limit approaching that point is a perfectly well-defined number.