Intuitive Notion of Limits

In mathematics, the concept of a limit provides a fundamental way to describe how a function behaves as its input approaches a particular value. Limits are essential in calculus and mathematical analysis, serving as the foundation for defining derivatives, integrals, and continuity. Intuitively, a limit captures the idea of getting arbitrarily close to a desired outcome, even if that outcome is never exactly reached.

To develop an intuitive understanding of limits, consider the example of a car approaching a stop sign. As the car moves closer, its distance to the sign decreases, eventually becoming almost negligible. Although the car never simultaneously occupies the position of the stop sign while still in motion, we can describe its approach using limits.

Consider the function f defined by