Understanding Limits
What Is a Limit?
A limit describes the value a function approaches as the input approaches a certain value. Formally, lim[x→c] f(x) = L means that f(x) gets arbitrarily close to L as x gets close to c. Limits are the foundation of calculus.
Left and Right Limits
The left-hand limit lim[x→c⁻] f(x) approaches c from below, while the right-hand limit lim[x→c⁺] f(x) approaches from above. A limit exists only when both one-sided limits exist and are equal.
When Limits Don't Exist
A limit may not exist when the left and right limits differ (jump discontinuity), when the function oscillates infinitely, or when the function grows without bound (approaches ±∞). These situations indicate important behavior of the function.
Epsilon-Delta Definition
The formal definition: lim[x→c] f(x) = L means for every ε > 0, there exists δ > 0 such that if 0 < |x − c| < δ, then |f(x) − L| < ε. This precise definition underpins all calculus proofs.
Applications
Limits define derivatives (as the limit of difference quotients) and integrals (as the limit of Riemann sums). They are used to analyze continuity, find asymptotes, evaluate indeterminate forms via L'Hôpital's rule, and study infinite series convergence.