Basic Concepts
In a nutshell: Limits tell us what value a function approaches; continuity means no breaks in the graph.
## Understanding Limits
Limits are the foundation of calculus. They describe how a function behaves as its input approaches a particular value, even if the function isn't actually defined at that point.
- If \( \lim_{x \to a} f(x) = L \), then as \( x \) gets closer to \( a \), \( f(x) \) gets closer to \( L \).
- Limits help define derivatives and integrals.
## Continuity
A function is continuous at a point if:
1. The function is defined at the point.
2. The limit exists at the point.
3. The value of the function equals the limit at that point.
## Why It Matters
The concept of limits allows us to work with functions that have jumps, holes, or even asymptotes, and is essential for understanding change.
Examples
- The limit of \( f(x) = \frac{\sin x}{x} \) as \( x \) approaches 0 is 1.
- A function with a hole at \( x = 2 \) is not continuous there.
Key terms
- Limit
- The value a function approaches as the input approaches a certain point.
- Continuity
- A property where a function has no breaks, jumps, or holes at a point.