Basic Concepts
Limits and Continuity
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:
- The function is defined at the point.
- The limit exists at the point.
- 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.
In a Nutshell
Limits tell us what value a function approaches; continuity means no breaks in the graph.