Definition: lim f(x) as x → a

A limit describes the value that a function approaches as the input variable approaches a specific point. It is the fundamental concept that makes calculus possible. Limits allow us to analyze what happens to a function near a certain value, even if the function is not defined exactly at that point. For example, when we write lim(x → a) f(x) = L, we mean that as x gets arbitrarily close to a, f(x) gets arbitrarily close to L. Limits help define continuity: a function is continuous if the limit at a point equals the function’s value at that point. Limits are also used to study infinite behavior, such as what happens when x approaches infinity. They allow us to define derivatives and integrals rigorously. In real life, limits help model processes that involve approximation, such as velocity at an exact moment in time. Without limits, we could not work with instantaneous rates of change.