Limits and Continuity

Calculus is often described as the mathematics of change, and it all begins with a single, deceptively simple question: what happens to a function's output as the input gets closer and closer to some target value? This question leads to the concept of a limit, the foundational idea upon which every other concept in calculus — derivatives, integrals, infinite series — is built. Without limits, none of those powerful tools would have a rigorous mathematical footing.

The beauty of limits is that they let us talk precisely about values a function approaches even when the function cannot actually reach that value. Consider the fraction $(x^2 - 4)/(x - 2)$. At $x = 2$ the denominator is zero, so the function is undefined there. Yet as $x$ creeps toward 2 from either side, the output creeps toward 4. The limit captures this approaching behavior and gives us a concrete number to work with, even at a point where the function itself has a hole.

Closely related to limits is the idea of continuity. Intuitively, a continuous function is one whose graph you can draw without lifting your pencil. Formally, continuity at a point requires three things: the function must be defined there, the limit must exist, and the two must agree. Most functions you encounter in everyday math — polynomials, exponentials, sine and cosine — are continuous everywhere on their domains. Discontinuities arise at points where something breaks: a division by zero, a jump between pieces, or a wild oscillation.

In this lesson you will learn how to evaluate limits using direct substitution, algebraic simplification, and factoring. You will also learn the formal definition of continuity and how to identify removable versus non-removable discontinuities. These skills form the gateway to everything that follows in calculus, so building a solid intuition here will pay dividends throughout the course.