1: Limits and Continuity
- Page ID
- 187310
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Calculus is built on two fundamental ideas:
- The derivative, which measures the instantaneous rate of change of a function.
- The integral, which measures the accumulated total of a quantity over an interval.
Both rely on a single underlying concept: the limit. Understanding limits enables us to describe how a function behaves near a point, even when the function is not explicitly defined at that point. Limits provide the foundation for derivatives and integrals, making them one of the most important ideas in calculus.
In this chapter, we focus on two closely related topics: limits and continuity. Together, they give us a precise language for describing when a function behaves smoothly, and when it does not.
Topics in this Chapter
- The Limit of a Function
- We introduce the definition of a limit, both intuitively and formally.
- Limits describe the value a function approaches, not necessarily the value it takes.
- This allows us to handle situations with jumps in a graph or asymptotes.
- Limit Laws
- Once we know what limits are, we need tools to compute them.
- The limit laws give us algebraic rules for combining limits, which make evaluating limits of complex functions much easier.
- Continuity
- Continuity captures the idea of a function with no “jumps,” “holes,” or “breaks.”
- Informally, a continuous function is one you can draw without lifting your pencil.
- Formally, continuity connects directly to the definition of limits.
Why this Matters
- Limits and continuity provide the rigorous foundation for both derivatives and integrals.
- They describe how functions behave near difficult or undefined points.
- They allow us to work with infinite processes and approximations.
In short: this chapter develops the basic language of calculus. Once you are comfortable with limits and continuity, you will be ready to explore differentiation and integration in the chapters that follow.