A limit describes what value a function approaches as the input gets closer and closer to a specific point — without necessarily reaching it. Limits are the foundation of calculus: derivatives and integrals are both defined using limits.
One-Sided Limits
The left-hand limit lim(x→a⁻) f(x) describes the value f approaches from the left. The right-hand limit lim(x→a⁺) f(x) describes the value from the right. For the two-sided limit to exist, both one-sided limits must be equal.
Types of Discontinuities
Removable discontinuity: the limit exists but f(a) is undefined or different (a "hole" in the graph). Jump discontinuity: left and right limits exist but are different. Infinite discontinuity: the function grows without bound near x = a (like 1/x at x = 0).
Why Limits Matter
The derivative is defined as a limit: f'(x) = lim(h→0) [f(x+h) − f(x)] / h. The definite integral is defined as a limit of Riemann sums. Without limits, neither concept would be rigorously defined.